Biderivations of the Algebra of Strictly Upper Triangular Matrices over a Commutative Ring
|
|
- Norah Blankenship
- 5 years ago
- Views:
Transcription
1 Journal of Mathematical Research & Exposition Nov., 2011, Vol. 31, No. 6, pp DOI: /j.issn: X Biderivations of the Algebra of Strictly Upper Triangular Matrices over a Commutative Ring Pei Sheng JI, Xiao Ling YANG, Jian Hui CHEN School of Mathematical Sciences, Qingdao University, Shandong , P. R. China Abstract Let N n(r) be the algebra consisting of all strictly upper triangular n n matrices over a commutative ring R with the identity. An R-bilinear map φ : N n(r) N n(r) N n(r) is called a biderivation if it is a derivation with respect to both arguments. In this paper, we define the notions of central biderivation and extremal biderivation of N n(r), and prove that any biderivation of N n(r) can be decomposed as a sum of an inner biderivation, central biderivation and extremal biderivation for n 5. Keywords biderivation; strictly upper triangular matrix; algebra. Document code A MR(2010) Subject Classification 17B30; 17B40 Chinese Library Classification O Introduction Let R be a commutative ring with the identity. Denote by N n (R), where n is a positive integer greater than 1, the R-algebra of strictly upper triangular n n matrices over R. In recent years, many significant researches have been done in automorphisms, Lie automorphisms, derivations and Lie derivations of N n (R). Cao [4] studied the R-algebra automorphisms of N n (R). Cao [5, 6] investigated the Lie automorphisms group of N n (R). Ou et al. [8] and Ji [7] determined the Lie derivations of N n (R). Wang and Li [9] studied the structure of all triple derivations of N n (R). Someone described the derivations of N n (R). Let T be an R-algebra and M a T-bimodule. Recall that an R-linear map d : T M is called a derivation if d(ab) = d(a)b + ad(b) for all a, b in T. For example, take x M, and set D x (a) = ax xa for all a in T. In the following, we denote by [x, y] the commutator (the Lie product) of the elements x, y T. A bilinear map φ : T T T is called a biderivation if it is a derivation with respect to both components, meaning that, φ(xy, z) = xφ(y, z) + φ(x, z)y and φ(x, yz) = φ(x, y)z + yφ(x, z) for all x, y, z T. If T is a noncommutative algebra, then the map φ(x, y) = λ[x, y] for all x, y T, where λ Z(T), the center of T, is a basic example of biderivation, and we call it an inner biderivation. Received April 8, 2010; Accepted May 28, 2010 Supported by the National Natural Science Foundation of China (Grant No ). * Corresponding author address: jipeish@yahoo.com (P. S. JI)
2 966 P. S. JI, X. L. YANG and J. H. CHEN Biderivations are a subject of research in various areas. Bresar et al. [3] proved that all biderivations on noncommutative prime rings are inner. Zhang et al. [10] showed that biderivations of nest algebras are usally inner. There are special cases of nest algebras with non-inner biderivations. Zhao et al. used the results in [10] to prove that every biderivation of an upper triangular matrix algebra is a sum of an inner biderivation and a special biderivation. Benkovic [1] generalized this result to triangular algebras. He proved that every biderivation of some certain triangular algebras is a sum of an inner biderivation and an extremal biderivation. The aim of this paper is to describe the biderivations of N n (R) for n 5. In the cases 2 n 4, we have already given the structure of the biderivations of N n (R). This paper is organized as follows. In Section 2, we construct certain special biderivation of N n (R) so as to build every biderivation of N n (R). In Section 3, we decompose any biderivation of N n (R) into a sum of those known ones. Now we introduce some preliminary notations and results about matrix algebras. Let e i,j be the standard matrix units for 1 i, j n. It is well known that the matrix set {e i,j : 1 i < j n} forms a basis of N n (R) and any x in N n (R) can be uniquely written as x = 1 i<j n x i,je i,j with x i,j R. Set M r = { x i,j e i,j N n (R) : x i,j R}, r = 1, 2,...,n 1, j i r and M r = 0 for r n. It is clear that each M r is an ideal of N n (R) and x 1 x 2 x r M r for any x 1, x 2,..., x r N n (R). It follows that M r M s M r+s. It is not difficult to know that the center of the R-algebra N n (R) is M = Re 1, To prove the main result of this paper, we need the following result, which might not be written down explicitly to our knowledge. Lemma 1.1 Let R be an arbitrary commutative ring with the identity and D a derivation of N n (R) for n 4. Then there is an n n diagonal matrix d, an element y N n (R) and c = (c 2,..., c ) R n 3 such that D(x) = [d, x] + [y, x] + i=2 x i,i+1c i e 1,n for all x = 1 i<j n x i,je i,j N n (R). 2. Certain standard biderivations of N n (R) In this section, we will give three types of standard derivations of N n (R), which will be used to describe arbitrary biderivations of N n (R). (A) Inner biderivations: Let d R. Then the map φ : N n (R) N n (R) N n (R), (x, y) d[x, y], is a biderivation of N n (R). We call it the inner biderivation of N n (R) induced by d. Note that the inner biderivation defined here is different from the one defined in the papers [1 3,10,11]. Since the center of N n (R) is Re 1,n, and for every x, y N n (R), we have that e 1,n [x, y] = 0. Hence the inner biderivation of N n (R) defined in [1 3,10,11] is the zero map. (B) Central biderivations: Let A be an (n 1) (n 1) matrix over R. We define an R- bilinear map φ A : N n (R) N n (R) N n (R) by φ A (y, x) = i,j=1 y i,i+1a i,j x j,j+1 e 1,n for all
3 Biderivations of the algebra of strictly upper triangular matrices over a commutative ring 967 x = 1 i<j n x i,je i,j, y = 1 i<j n y i,je i,j N n (R). Then it is easy to check that φ A is a biderivation of N n (R). We call it the central biderivation of N n (R) induced by A. (C) Extremal biderivations: Let a R. The R-bilinear map φ a : N n (R) N n (R) N n (R), which is defined by φ a (y, x) = a(y 2,3 x 1,3 e 1,n + y 2,3 x 2,3 e 2,n + y 1,3 x 2,3 e 1,n ) for all x = 1 i<j n x i,je i,j, y = 1 i<j n y i,je i,j N n (R), is a biderivation on N n (R). In fact, fix y = 1 i<j n y i,je i,j N n (R), for all x = 1 i<j n x i,je i,j, x = 1 i<j n x i,j e i,j, since (xx ) 2,3 = 0 and (xx ) 1,3 = x 1,2 x 2,3, we have that φ a(y, xx ) = ay 2,3 x 1,2 x 2,3 e 1,n. Computing directly, we get that φ a (y, x)x + xφ a (y, x ) =a(y 2,3 x 1,3 e 1,n + y 2,3 x 2,3 e 2,n + y 1,3 x 2,3 e 1,n )x + ax(y 2,3 x 1,3e 1,n + y 2,3 x 2,3e 2,n + y 1,3 x 2,3e 1,n ) =ax 1,2 y 2,3 x 2,3 e 1,n. Thus φ a (y, xx ) = φ a (y, x)x + xφ a (y, x ). This means that φ a is a biderivation with respect to the second component. Since φ a is a symmetric bilinear map, we have that φ a is a derivation with respect to the first component. Therefore φ a is a biderivation of N n (R). We call it the extremal biderivation of N n (R) induced by a R. 3. Biderivation of N n (R) In this section, we give the main result of this paper. Theorem 3.1 Let R be an arbitrary commutative ring with the identity and φ a biderivation of N n (R). If n 5, then there are elements a, d R and an (n 1) (n 1) matrix A over R such that for all x, y N n (R). φ(y, x) = d[y, x] + φ A (y, x) + φ a (y, x) In order to prove our main result, we need the following lemmas. Lemma 3.2 ([2, Corollary 2.4]) Let T be an algebra on a commutative ring R and φ : T T T a biderivation. Then for all x, y, u, v A. φ(x, y)[u, v] = [x, y]φ(u, v) In what follows, for 1 i < j n, since φ(e i,j, x) : N n (R) N n (R) is a derivation of N n (R), by Lemma 1.1, we suppose that φ(e i,j, x) = [d i,j, x] + [y i,j, x] + c i,j i x i,i+1 e 1,n for all x = 1 i<j n x i,je i,j N n (R), where d i,j = diag(d i,j 1,..., di,j n ) is a diagonal matrix over R, y i,j = 1 t<s n yi,j t,se i,j N n (R), and (c i,j 2,...,ci,j ) Rn 3. Lemma 3.3 For any (i, j) (1 i < j n), there is d i,j R such that d i,j = diag(d i,j, d i,j,...,d i,j ). i=2
4 968 P. S. JI, X. L. YANG and J. H. CHEN Hence [d i,j, x] = 0 for all x N n (R). Proof Step 1. First we will prove that if there is e t,s with s t < n 2 such that [e i,j, e t,s ] = 0, then d i,j t = d i,j s. Since [e i,j, e t,s ] = 0, by Lemma 3.2, we get that Hence φ(e i,j, e t,s )[u, v] = [e i,j, e t,s ]φ(u, v) = 0 for all u, v N n (R). φ(e i,j, e t,s ) =[d i,j, e t,s ] + [y i,j, e t,s ] + c i,j t δ t+1,s e 1,n =(d i,j t d i,j s )e t,s + [y i,j, e t,s ] + c i,j t δ t+1,s e 1,n M, where δ is the Kronecker s delta. Since s t < n 2, e t,s M. Since e t,s M s t \M s t+1 and [y i,j, e t,s ] + c i,j t δ t+1,s e 1,n M s t+1, we have d i,j t d i,j s = 0, i.e., d i,j t = d i,j s. Henceforth fix (i, j) (1 i < j n). Step 2. (i) Clearly, [e i,j, e 1,k ] = 0 for each 1 < k n 2 with k i. It follows from step 1 that d i,j 1 = d i,j k for all 1 < k n 2 with k i. (ii) Since [e i,j, e k,n ] = 0 for all 2 < k n 1 with k j, by Step 1, we have that d i,j k = di,j n for all 2 < k n 1 with k j. (iii) Since [e i,j, e i,k ] = 0 and [e i,j, e t,j ] = 0, by Step 1, we get that d i,j i i < k min{n, n 3 + i} and d i,j j = d i,j t all min{i, max{0, j n + 2}} k max{j, min{n, n 3 + i}}. Case 1 i = 1. = d i,j k for all j > k max{0, j n + 2}. Thus d i,j i = d i,j k It follows from Step 2 (i) that d 1,j 1 = = d 1,j. If j < n 2, by Step 2 (ii), we have that d 1,j = d1,j = d1,j So d 1,j 1 = d 1,j 2 = = d 1,j If j = n 2, then, by Step 2 (ii), d 1, suffices to prove that d 1, 2 = d 1, 1, we get d 1, 2 = d 1,. Thus d1, = d 1, n for all for. In this case, to complete the proof, it. Since [e 1,, e 2, ] = 0 and (n 1) 2 < n 2, by Step 1 = d 1, 2 = = d 1, If j = n 1, then, by Step 2 (ii), d 1, = d 1, Since [e 1,, e, ] = 0 and (n 1) (n 2) < n 2, by Step 1, we get d 1, = d 1,. Thus d1,1 = d 1, 2 = = d 1, When j = n, it follows from Step (ii) that d 1,n 3 = d 1,n 4 = = d 1,n Thus d 1,n 1 = d 1,n 2 = = d 1,n Case 2 i = 2. It follows from Step 2 (i) that d 2,j 1 = d 2,j 3 = = d 2,j, and from Step 2 (iii) that d2,j d 2,j 3 = = d 2,j. Thus d2,j 1 = d 2,j 2 = = d 2,j. If j n 2, by Step 2 (ii), we have that d 2,j d 2,n If j = n 1, then, by Step 2 (ii), d 1, = d2,j n = d 1, n When j = n, it follows from Step (ii) that d 2,n 3 = d 2,n. So d2,j 1 = d 2,j 2 = = d 2,j 1 = d 2, 2 =. Thus d 2, 2 = = d 2, 4 = = d 2,n Thus d2,n 1 = d 2,n 2 = =
5 Biderivations of the algebra of strictly upper triangular matrices over a commutative ring 969 Case 3 i 3. It follows from Step 2 (i) that d i,j 1 = d i,j 2 = = d i,j i 1, and from Step 2 (iii) that di,ji = d i,j i+1 = = di,j Since [e i,j, e i 1,i+1 ] = 0 and (i + 1) (i 1) < n 2, by Step 1, we get d i,j i 1 = di,j i+1. Thus di,j 1 = d i,j 2 = = d i,j This completes the proof. Henceforth, for 1 i < j n, we suppose that φ(e i,j, x) = [y i,j, x] + c i,j i x i,i+1 e 1,n for all x = 1 i<j n x i,je i,j N n (R), where y i,j = 1 t<s n yi,j t,se i,j N n (R), and (c i,j 2,...,ci,j ) R n 3. Since [e 1,n, x] = 0 for all x N n (R), we suppose that y i,j 1,n = 0 for all (i, j). In the proof of the following lemmas, we often use the following fact: Suppose c i,j R for all 1 i < j n. If 1 i<j n c i,jx i,j = 0 for all x = 1 i<j n x i,je i,j N n (R), then c i,j = 0 for all 1 i < j n. And for every z N n (R), and i=2 φ(e i,j, x)z = ([y i,j, x] + c i,j i x i,i+1 e 1,n )z = [y i,j, x]z i=2 zφ(e i,j, x) = z([y i,j, x] + c i,j i x i,i+1 e 1,n ) = z[y i,j, x]. Lemma 3.4 There exists an element d R such that (i) y 1,j = de 1,j + y 1,j 1, e 1, + y 1,j 2,n e 2,n for j < n 1; (ii) y 1,n = y 1,n 1, e 1, + y 1,n 2,n e 2,n; (iii) y 1, = y 1, 1, e 1, + y 1, 1, e 1, + y 1, 2,n e 2,n ; (iv) y i,j = de i,j + y i,j 1, e 1, + y i,j 2,n e 2,n for j n 1 and i > 2; (v) y i, = y i, 1, e 1, + y i, 1, e 1, + y 1, 2,n e 2,n + de i, for i > 2. i=2 Proof (I) For 1 < j n, it follows from e 1,2 e 1,j = 0 that 0 = φ(e 1,2 e 1,j, x) = e 1,2 φ(e 1,j, x) + φ(e 1,2, x)e 1,j = e 1,2 φ(e 1,j, x) = e 1,2 [y 1,j, x] n = ( y 1,j 2,t x t,s x 2,t yt,s)e 1,j 1,s s=4 2<t<s 2<t<s for all x = 1 i<j n x i,je i,j N n (R). Hence, for every 4 s n, 2<t<s y1,j 2,t x t,s 2<t<s x 2,ty 1,j t,s = 0 for all x = 1 i<j n x i,je i,j N n (R), which implies that y 1,j t,s = 0 for t > 2 and y 1,j 2,t = 0 for 3 t n 1. So y1,j = y1,j 1,t e 1,t + y 1,j 2,n e 2,n. (II) For j < n 1, since e 1,j e,n = 0, by (I), we have that 0 = φ(e 1,j e,n, x) = e 1,j φ(e,n, x) + φ(e 1,j, x)e,n = e 1,j [y,n, x] + [y 1,j, x]e,n n = ( y,n j,t x t,s x j,t y,n t,s )e 1,s + y 1,j 1,t x t,e 1,n s=j+2 j<t<s j<t<s for all x = 1 i<j n x i,je i,j N n (R). Therefore j<t<n y,n j,t x t,n j<t<s x j,ty,n t,n + 1,t x t, = 0 and j<t<s y,n x t,s j<t<s x j,ty,n t,s = 0 for j + 2 s < n, which y1,j j,t
6 970 P. S. JI, X. L. YANG and J. H. CHEN implies that y 1,j 1,t = y,n for all j < n 1, and y1,j 1,t = 0 for 2 t n 2 with t j. Suppose d = y,n, We have that y 1,j = de 1,j + y 1,j 1, e 1, + y 1,j 2,n e 2,n for j < n 1. (i) is proved. (III) Since e 1,n e,n = 0, by (I), we get that 0 = φ(e 1,n e,n, x) = e 1,n φ(e,n, x) + φ(e 1,n, x)e,n = φ(e 1,n, x)e,n = [y 1,n, x]e,n = y 1,n 1,t x t,e 1,n for all x = 1 i<j n x i,je i,j N n (R). Hence we have that y 1,n 1,t y 1,n = y 1,n 1, e 1, + y 1,n 2,n e 2,n. (IV) For 2 < i < n, since e 1,2 e i,j = 0, by (i), we have that = 0 for all 2 t n 2. Thus 0 = φ(e 1,2 e i,j, x) = e 1,2 φ(e i,j, x) + φ(e 1,2, x)e i,j = e 1,2 [y i,j, x] + [y 1,2, x]e i,j n = ( y i,j 2,t x t,s x 2,t yt,s)e i,j 1,s + dx 2,i e 1,j s=4 2<t<s 2<t<s for all x = 1 i<j n x i,je i,j N n (R). These imply that, for every 4 s n with s j, 2<t<s yi,j 2,t x t,s 2<t<s x 2,ty i,j t,s = 0, and that 2<t<j yi,j 2,t x t,j 2<t<j x 2,ty i,j t,j + dx 2,i = 0 for all x = 1 i<j n x i,je i,j N n (R). Therefore y i,j i,j = d and yi,j t,s = 0 unless t = 1 and (t, s) = (i, j), (2, n). So y i,j = yi,j 1,t e 1,t + y i,j 2,n e 2,n + de i,j. (V) For any (i, j) (i > 2 and j < n 1), since e i,j e,n = 0, by (IV), we have that 0 = φ(e i,j e,n, x) = e i,j φ(e,n, x) + φ(e i,j, x)e,n = e i,j [y,n, x] + [y i,j, x]e,n = dx j, e i,n + y i,j 1,t x t,e 1,n + dx j, e i,n = y i,j 1,t x t,e 1,n for all x = 1 i<j n x i,je i,j N n (R). This leads to yi,j 1,t x t, = 0, which implies that y i,j 1,t = 0 for all 2 t n 2. Thus yi,j = de i,j + y i,j 1, e 1, + y i,j 2,n e 2,n for j < n 1. (VI) For any i (i > 2), by (IV), it follows from e i,n e,n = 0 that 0 = φ(e i,n e,n, x) = e i,n φ(e,n, x) + φ(e i,n, x)e,n = φ(e i,n, x)e,n = [ y i,n 1,t e 1,t + y i,n 2,n e 2,n + de i,n, x]e,n = y i,n 1,t x t,e 1,n for all x = 1 i<j n x i,je i,j N n (R). Hence y i,n 1,t = 0 for all 2 t n 2. So y i,n = y i,n 1, e 1, + y i,n 2,n e 2,n + de i, (VII) Since e 1, e,n = 0, using (I) and (VI), we can get that 0 = φ(e 1, e,n, x) = e 1, φ(e,n, x) + φ(e 1,, x)e,n = e 1, [y,n 1, e 1, + y,n n 3 = y 1, 1,t x t, e 1,n 2,n e 2,n + de,n, x] + [ y 1, 1,t e 1,t + y 1, 2,n e 2,n, x]e,n
7 Biderivations of the algebra of strictly upper triangular matrices over a commutative ring 971 for all x = 1 i<j n x i,je i,j N n (R). Therefore n 3 x = 1 i<j n y1, x i,j e i,j N n (R), which implies that y 1, 1,t = 0 for all 2 t n 3. Thus 1,t x t, = 0 for all y 1, = y 1, 1, e 1, + y 1, 1, e 1, + y 1, 2,n e 2, (VIII) For i > 2, since e i, e,n = 0, by (VI), we have that 0 = φ(e i, e,n, x) = e i, φ(e,n, x) + φ(e i,, x)e,n n 3 = y i, 1,t x t, e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore n 3 x = 1 i<j n yi, x i,j e i,j N n (R), which leads to that y 1, 1,t = 0 for all 2 t n 3. So 1,t x t, = 0 for all y i, = y i, 1, e 1, + y i, 1, e 1, + y 1, 2,n e 2,n + de i,. Lemma 3.5 For i = 2, we have the following equalities: (i) y 2,j = de 2,j + y 2,j 1, e 1, + y 2,j 3,n e 3,n + y 2,j 2,n e 2,n for j < n 1; (ii) y 2, = y 2, 1, e 1, + y 2, 1, e 1, + y 2, 3,n e 3,n + y 2, 2, e 2, + y 2, 2,n e 2,n ; (iii) y 2,n = y 2,n 1, e 1, + y 2,n 2,n e 2,n + y 2,n 2, e 2, + y 2,n 3,n e 3,n. Proof (I) For any j > 2, since e 1,3 e 2,j = 0, by Lemma 3.4(i), we have that 0 = φ(e 1,3 e 2,j, x) = e 1,3 φ(e 2,j, x) + φ(e 1,3, x)e 2,j = e 1,3 [y 2,j, x] + [y 1,3, x]e 2,j n = ( y 2,j 3,t x t,s s=5 3<t<s 3<t<s x 3,t y 2,j t,s )e 1,s for all x = 1 i<j n x i,je i,j N n (R). Hence, for every 5 s n, 3<t<s y2,j 3,t x t,s 3<t<s x 3,ty 2,j t,s = 0 for all x = 1 i<j n x i,je i,j N n (R). Therefore y 2,j t,s = 0 for t > 3 and y 2,j 3,t = 0 for 4 t n 1. So y2,j = y2,j 1,t e 1,t + n t=3 y2,j 2,t e 2,t + y 2,j 3,n e 3,n. (II) For 3 j < n 1, using (I) and Lemma 3.4 (iv), it follows from e 2,j e,n = 0 that 0 = φ(e 2,j e,n, x) = e 2,j φ(e,n, x) + φ(e 2,j, x)e,n = e 2,j [y,n, x] + [y 2,j, x]e,n = dx j, e 2,n + y 2,j 1,t x t,e 1,n + t=3 y 2,j 2,t x t,e 2,n y 2,j 2, x 1,2e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore y2,j 1,t x t, y 2,j 2, x 1,2 = 0 and dx j, + t=3 y2,j 2,t x t, = 0 for all x = 1 i<j n x i,je i,j N n (R). Hence y 2,j 1,t = 0 for all 2 t
8 972 P. S. JI, X. L. YANG and J. H. CHEN n 2, d = y 2,j 2,j, and y2,j 2,t = 0 for 2 < t n 1 with t j. Thus y2,j = de 2,j + y 2,j 1, e 1, + y 2,j 3,n e 3,n + y 2,j 2,n e 2,n for j < n 1. (III) Since e 2, e,n = 0, by (I) and Lemma 3.4(iv), we have that 0 =φ(e 2, e,n, x) = e 2, φ(e,n, x) + φ(e 2,, x)e,n =e 2, [y,n 1, e 1, + y,n 2,n e 2,n + de,n, x]+ n [ n 3 = t=3 y 2, 1,t e 1,t + t=3 n 3 y 2, 2,t x t, e 2,n + y 2, 2,t e 2,t + y 2, 3,n e 3,n, x]e,n y 2, 1,t x t, e 1,n y 2, 2, x 1,2e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore n 3 y2, 1,t x t, y 2, 2, x 1,2 = 0 and n 3 t=3 y2, 2,t x t, = 0 for all x = 1 i<j n x i,je i,j N n (R). It follows that y 2, 1,t = 0 for all 2 t n 3 and y 2, 2,t = 0 for all 3 t n 2. Thus y 2, = y 2, 1, e 1, + y 2, 1, e 1, + y 2, 3,n e 3,n + y 2, 2, e 2, + y 2, 2,n e 2, (IV) Since e 2,n e,n = 0, by (I), we have that 0 = φ(e 2,n e,n, x) = e 2,n φ(e,n, x) + φ(e 2,n, x)e,n n = φ(e 2,n, x)e,n = [ y 2,n 1,t e 1,t + y 2,n 2,t e 2,t + y 2,n 3,n e 3,n, x]e,n = y 2,n 1,t x t,e 1,n + t=3 t=3 y 2,n 2,t x t,e 2,n y 2,n 2, x 1,2e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore y2,n 1,t x t, y 2,n 2, x 1,2 = 0 and t=3 y2,n 2,t x t, = 0 for all x = 1 i<j n x i,je i,j N n (R), which implies that y 2,n 1,t = 0 for 2 t n 2 and y 2,n 2,t = 0 for all 3 t n 1. Thus y 2,n = y 2,n 1, e 1, + y 2,n 2,n e 2,n + y 2,n 2, e 2, + y 2,n 3,n e 3,n. Lemma 3.6 (i) For all x N n (R), we have φ(e 1,n, x) = 0; (ii) For 3 < j < n, we get y 1,j = de 1,j and c 1,j t = 0 for 2 t n 2; (iii) Then y 1,3 = de 1,3, c 1,j t = 0 for 3 t n 2 and c 1,3 2 = y 2,3 3, Proof (I) By Lemma 3.4 (i) and (iv), it is easy to check that for all x N n (R). φ(e 1,n, x) =φ(e 1,3 e 3,n, x) = e 1,3 φ(e 3,n, x) + φ(e 1,3, x)e 3,n =e 1,3 [de 3,n + y 3,n 1, e 1, + y 3,n 2,n e 2,n, x]+ =0 [de 1,3 + y 1,3 1, e 1, + y 1,3 2,n e 2,n, x]e 3,n (II) Fix any i (2 < i < n 1), since φ(e 1,, x) = φ(e 1,i e i,, x) = e 1,i φ(e i,, x) +
9 Biderivations of the algebra of strictly upper triangular matrices over a commutative ring 973 φ(e 1,i, x)e i,, using Lemma 3.4 (i) and (v), we have that i.e., [y 1, 1, e 1, + y 1, 1, e 1, + y 1, 2,n e 2,n, x] + c 1, t x t,t+1 e 1,n = e 1,i [y i, 1, e 1, + y i, 1, e 1, + y 1, 2,n e 2,n + de i,, x]+ [de 1,i + y 1,i 1, e 1, + y 1,i 2,n e 2,n, x]e i,, y 1, 1, x,e 1, + y 1, 1, x,ne 1,n + y 1, 1, x,ne 1,n y 1, 2,n x 1,2 e 1,n + c 1, t x t,t+1 e 1,n = dx,n e 1,n for all x = 1 i<j n x i,je i,j N n (R). Hence y 1, 1, x, = 0 and y 1, 1, x,n + y 1, 1, x,n y 1, 2,n x 1,2 + c 1, t x t,t+1 = dx,n for all x = 1 i<j n x i,je i,j N n (R). Therefore y 1, 1, = 0, y1, 2,n = 0, y 1, 1, = d and c 1, t = 0 for 2 t n 2. So y 1, = de 1,. (III) For 4 j n 2, since φ(e 1,j, x) = φ(e 1,3 e 3,j, x) = e 1,3 φ(e 3,j, x) + φ(e 1,3, x)e 3,j, we get, using Lemma 3.4 (i, iv), that y 1,j 1, x,ne 1,n y 1,j 2,n x 1,2e 1,n + de 1,j x + c 1,j t x t,t+1 e 1,n = de 1,j x for all x = 1 i<j n x i,je i,j N n (R). Hence y 1,j 1, x,n y 1,j 2,n x 1,2 + c1,j t x t,t+1 = 0 for all x = 1 i<j n x i,je i,j N n (R), which implies that y 1,j 1, = y1,j 2,n 2 t n 2. Thus y 1,j = de 1,j. = 0 and c1,j t = 0 for (IV) Since φ(e 1,3, x) = φ(e 1,2 e 2,3, x) = e 1,2 φ(e 2,3, x)+φ(e 1,2, x)e 2,3, we see using Lemma 3.4 (i) and Lemma 3.5 (i) that y 1,3 1, x,ne 1,n y 1,3 2,n x 1,2e 1,n + de 1,3 x + c 1,3 t x t,t+1 e 1,n = de 1,3 x y 2,3 3,n x 2,3e 1,n for all x = 1 i<j n x i,je i,j N n (R). Hence y 1,3 1, x,n y 1,3 2,n x 1,2 + c1,3 t x t,t+1 = y 2,3 3,n x 2,3 for all x = 1 i<j n x i,je i,j N n (R). Therefore y 1,3 1, = y1,3 2,n = 0 for 3 t n 2. So y 1,3 = de 1,3. c 1,3 t Lemma 3.7 For 3 < j n, we get that y 2,j = de 2,j and c 2,j t = 0 for 2 t n 2. = 0, c1,3 2 = y 2,3 3,n and Proof (I) Since φ(e 2,n, x) = φ(e 2,3 e 3,n, x) = e 2,3 φ(e 3,n, x) + φ(e 2,3, x)e 3,n, by Lemma 3.4 (iv) and Lemma 3.5 (i, iii), we have that y 2,n 1, x,ne 1,n y 2,n 2,n x 1,2e 1,n y 2,n 2, x 1,2e 1, + y 2,n 2, x,ne 2,n y 2,n 3,n x 1,3e 1,n y 2,n 3,n x 2,3e 2,n + c 2,n t x t,t+1 e 1,n
10 974 P. S. JI, X. L. YANG and J. H. CHEN = dx 1,2 e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore y 2,n 2, x 1,2 = 0, y 2,n 2, x,n y 2,n 3,n x 2,3 = 0 and y 2,n 1, x,n y 2,n 2,n x 1,2 y 2,n 3,n x 1,3+ t x t,t+1 = dx 1,2 for all x = 1 i<j n x i,je i,j N n (R), which means that y 2,n 2,n Thus y 2,n = de 2, c2,n = d, y2,n 1, = y2,n 2, = y2,n 3,n = 0, and c2,n t = 0 for 2 t n 2. (II) Since φ(e 2,, x) = φ(e 2,3 e 3,, x) = e 2,3 φ(e 3,, x) + φ(e 2,3, x)e 3,, by Lemma 3.4 (v) and Lemma 3.5 (i, ii), we have that y 2, 1, x,e 1, + y 2, 1, x,ne 1,n + y 2, 1, x,ne 1,n + y 2, 2, x,ne 2,n y 2, 2, x 1,2e 1, y 2, 2,n x 1,2 e 1,n y 2, 3,n x 1,3 e 1,n y 2, 3,n x 2,3 e 2,n + c 2, t x t,t+1 e 1,n = dx,n e 2,n dx 1,2 e 1, for all x = 1 i<j n x i,je i,j N n (R). Therefore y 2, 2, x,n y 2, 3,n x 2,3 = dx,n, y 2, 1, x, y 2, 2, x 1,2 = dx 1,2, y 2, 1, x,n + y 2, 1, x,n y 2, 2,n x 1,2 y 2, 3,n x 1,3 + c 2, t x t,t+1 e 1,n = 0 for all x = 1 i<j n x i,je i,j N n (R). This implies that y 2, 3,n = y 2, 1, = 0, y2, 2, = d, y 2, 1, = y2, 2,n = 0, and c 2, t = 0 for 2 t n 2. So y 2, = de 2,. (III) For 4 j < n 1, since φ(e 2,j, x) = φ(e 2,3 e 3,j, x) = e 2,3 φ(e 3,j, x) + φ(e 2,3, x)e 3,j, using Lemma 3.4 (iv) and Lemma 3.5 (i), we can get that y 2,j 1, x,ne 1,n y 2,j 2,n x 1,2e 1,n dx 1,2 e 1,j + de 2,j x y 2,j 3,n x 1,3e 1,n y 2,j 3,n x 2,3e 2,n + c 2,j t x t,t+1 e 1,n = de 2,j x dx 1,2 e 1,j for all x = 1 i<j n x i,je i,j N n (R). Therefore y 2,j 1, x,n y 2,j 2,n x 1,2 y 2,j 3,n x 1,3 + c 2,j t x t,t+1 = 0 for all x = 1 i<j n x i,je i,j N n (R). Hence y 2,j 1, = y2,j 2,n = y2,j 3,n 2 t n 2. So y 2,j = de 2,j. = 0, and c2,j t = 0 for Lemma 3.8 For i > 2 and j i 2, we have that y i,j = de i,j and c 2,j t = 0 for 2 t n 2. Proof (I) Since φ(e i,n, x) = φ(e i, e,n, x) = e i, φ(e,n, x) + φ(e i,, x)e,n, by Lemma 3.4 (iv) and (v), we have that [y i,n 1, e 1, + y i,n 2,n e 2,n + de i,n, x] + c i,n t x t,t+1 e 1,n
11 Biderivations of the algebra of strictly upper triangular matrices over a commutative ring 975 i.e., = e i, [de,n + y,n 1, e 1, + y,n 2,n e 2,n, x]+ [de i, + y i, 1, e 1, + y i, 2,n e 2,n + y i, 1, e 1,, x]e,n, dxe i,n + y i,n 1, x,ne 1,n y i,n 2,n x 1,2e 1,n + c i,n t x t,t+1 e 1,n = dxe i,n + y i, 1, x,e 1,n for all x = 1 i<j n x i,je i,j N n (R). Therefore y i,n 1, x,n y i,n 2,n x 1,2 + c i,n t x t,t+1 = y i, 1, x, for all x = 1 i<j n x i,je i,j N n (R). Hence y i,n 1, = yi,n 2,n = 0, yi, 1, = ci,n and c2,n t = 0 for 2 t n 3. Thus y i,n = de i, (II) Since φ(e i,, x) = φ(e i, e,, x) = e i, φ(e,, x) + φ(e i,, x)e, by Lemma 3.4 (iv) and (v), we have that dx,n e i,n dxe i, + y i, 1, x,ne 1,n y i, 2,n x 1,2e 1,n + y i, 1, x,e 1, + y i, 1, x,ne 1,n + c i, t x t,t+1 e 1,n = dx,n e i,n dxe i, for all x = 1 i<j n x i,je i,j N n (R). Therefore y i, 1, x, = 0, and y i, 1, x,n y i, 2,n x 1,2 + y i, 1, x,n + ci, t x t,t+1 = 0 for all x = 1 i<j n x i,je i,j N n (R). This implies that y i, 1, = yi, 1, = yi, 2,n = 0, and c 2, t = 0 for 2 t n 2. Thus y i, = de i,. By (I), we have that c i,n = yi, 1, = 0. (III) For any (i, j) (i > 2, i + 2 < j < n 1), since φ(e i,j, x) = φ(e i,i+1 e i+1,j, x) = e i,i+1 φ(e i+1,j, x) + φ(e i,i+1, x)e i+1,j, using Lemma 3.4 (iv), we can get that de i,j x dxe i,j + y i,j 1, x,ne 1,n y i,j 2,n x 1,2e 1,n + c i,j t x t,t+1 e 1,n = de i,j x dxe i,j for all x = 1 i<j n x i,je i,j N n (R). Therefore y i,j 1, x,n y i,j 2,n x 1,2 + ci,j t x t,t+1 = 0 for all x = 1 i<j n x i,je i,j N n (R). Hence y i,j 1, = yi,j 2,n = 0, and ci,j t = 0 for 2 t n 2. So y i,j = de i,j. Summarizing Lemmas 3.4 (i), 3.5 (i) and , we have the following result. Corollary 3.9 The following equalities hold (i) φ(e 1,n, x) = 0 for all x = 1 i<j n x i,je i,j N n (R); (ii) For 1 i n 1 with i 2, φ(e i,i+1, x) = [de i,i+1 + y i,i+1 1, e 1, + y i,i+1 2,n e 2,n, x] + x t,t+1 e 1,n for all x N n (R); ci,i+1 t (iii) For i + 2 j, except (i, j) = (1, 3), φ(e i,j, x) = [de i,j, x] for all x N n (R); (iv) φ(e 1,3, x) = [de 1,3, x] + c 1,3 2 x 2,3e 1,n for all x N n (R); (v) φ(e 2,3, x) = [de 2,3 + y 2,3 1, e 1, + y 2,3 2,n e 2,n + y 2,3 3,n e 3,n, x] + ci,i+1 t x t,t+1 e 1,n for all x N n (R), and y 2,3 3,n = c1,3 2.
12 976 P. S. JI, X. L. YANG and J. H. CHEN The Proof of Theorem 3.1 For 1 i n 1, denote y i,i+1 1, by ci,i+1, yi,i+1 2,n by c i,i+1 1, and c 1,3 2 = y 2,3 3,n by a. Then, for i 2, and for all x N n (R). Let A denote φ(e i,i+1, x) = [de i,i+1, x] + c i,i+1 t x t,t+1 e 1,n, t=1 φ(e 2,3, x) = [de 2,3, x] + [ ae 3,n, x] + c i,i+1 t x t,t+1 e 1,n, t=1 φ(e 1,3, x) = [de 1,3, x] + ax 2,3 e 1,n i,j=1 ci,i+1 j N n (R), it follows from Corollary 3.9 that φ(y, x) = y i,j φ(e i,j, x) 1 i<j n = 1 i<j n y i,j [de i,j, x] + e i,j. For y = 1 i<j n y i,je i,j, x = 1 i<j n x i,je i,j i=1 (y i,i+1 j=1 c i,i+1 j x j,j+1 e 1,n )+ y 2,3 [ ae 3,n, x] + ay 1,3 x 2,3 e 1,n =d[ y i,j e i,j, x] + (y i,i+1 c i,i+1 j x j,j+1 e 1,n )+ 1 i<j n i=1 j=1 a(y 2,3 x 1,3 e 1,n + y 2,3 x 2,3 e 2,n + y 1,3 x 2,3 e 1,n ) =d[y, x] + φ A (y, x) + φ a (y, x). We thus complete the proof of Theorem 3.1. Acknowledgments The authors would like to thank the referees for their comments. References [1] BENKOVIČ D. Biderivations of triangular algebras [J]. Linear Algebra Appl., 2009, 431(9): [2] BREŠAR B. On generalized biderivations and related maps [J]. J. Algebra, 1995, 172(3): [3] BREŠAR B, MARTINDALE W S, MIERS C R. Centralizing maps in prime rings with involution [J]. J. Algebra, 1993, 161(2): [4] CAO You an, WANG Jingtong. A note on algebra automorphisms of strictly upper triangular matrices over commutative rings [J]. Linear Algebra Appl., 2000, 311(1-3): [5] CAO You an. Automorphisms of the Lie algebra of strictly upper triangular matrices over certain commutative rings [J]. Linear Algebra Appl., 2001, 329(1-3): [6] CAO You an, TAN Zuowen. Automorphisms of the Lie algebra of strictly upper triangular matrices over a commutative ring [J]. Linear Algebra Appl., 2003, 360: [7] JI Peisheng, YUAN Huali. Lie derivations of strictly upper triangular matrices over commutative rings [J]. Acta Math. Sinica (Chin. Ser.), 2007, 50(4): (in Chinese) [8] OU Shikun, WANG Dengyin, YAO Ruiping. Derivations of the Lie algebra of strictly upper triangular matrices over a commutative ring [J]. Linear Algebra Appl., 2007, 424(2-3): [9] WANG Hengtai, LI Qingguo. Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring [J]. Linear Algebra Appl., 2009, 430(1): [10] ZHANG Jianhua, FENG Shan, LI Hongxia, et al. Generalized biderivations of nest algebras [J]. Linear Algebra Appl., 2006, 418(1):
Product Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras
Journal of Mathematical Research with Applications Sept., 2013, Vol.33, No. 5, pp. 528 542 DOI:10.3770/j.issn:2095-2651.2013.05.002 Http://jmre.dlut.edu.cn Product Zero Derivations on Strictly Upper Triangular
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 435 (2011) 2889 2895 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Idempotent elements
More informationBRUNO L. M. FERREIRA AND HENRIQUE GUZZO JR.
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 60, No. 1, 2019, Pages 9 20 Published online: February 11, 2019 https://doi.org/10.33044/revuma.v60n1a02 LIE n-multiplicative MAPPINGS ON TRIANGULAR n-matrix
More informationPERMANENTLY WEAK AMENABILITY OF REES SEMIGROUP ALGEBRAS. Corresponding author: jabbari 1. Introduction
International Journal of Analysis and Applications Volume 16, Number 1 (2018), 117-124 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-117 PERMANENTLY WEAK AMENABILITY OF REES SEMIGROUP
More informationAdditivity Of Jordan (Triple) Derivations On Rings
Fayetteville State University DigitalCommons@Fayetteville State University Math and Computer Science Working Papers College of Arts and Sciences 2-1-2011 Additivity Of Jordan (Triple) Derivations On Rings
More informationSince Brešar and Vukman initiated the study of left derivations in noncom-
JORDAN LEFT DERIVATIONS IN FULL AND UPPER TRIANGULAR MATRIX RINGS XIAO WEI XU AND HONG YING ZHANG Abstract. In this paper, left derivations and Jordan left derivations in full and upper triangular matrix
More informationGroup, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,
Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ
More informationAdditivity of maps on generalized matrix algebras
Electronic Journal of Linear Algebra Volume 22 Volume 22 (2011) Article 49 2011 Additivity of maps on generalized matrix algebras Yanbo Li Zhankui Xiao Follow this and additional works at: http://repository.uwyo.edu/ela
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 437 (2012) 2719 2726 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Lie derivations
More informationRational constants of monomial derivations
Rational constants of monomial derivations Andrzej Nowicki and Janusz Zieliński N. Copernicus University, Faculty of Mathematics and Computer Science Toruń, Poland Abstract We present some general properties
More informationStrongly Nil -Clean Rings
Strongly Nil -Clean Rings Abdullah HARMANCI Huanyin CHEN and A. Çiğdem ÖZCAN Abstract A -ring R is called strongly nil -clean if every element of R is the sum of a projection and a nilpotent element that
More informationOn zero product determined algebras
Linear and Multilinear Algebra ISSN: 0308-1087 (Print) 1563-5139 (Online) Journal homepage: http://www.tandfonline.com/loi/glma20 On zero product determined algebras Daniel Brice & Huajun Huang To cite
More informationOn Regularity of Incline Matrices
International Journal of Algebra, Vol. 5, 2011, no. 19, 909-924 On Regularity of Incline Matrices A. R. Meenakshi and P. Shakila Banu Department of Mathematics Karpagam University Coimbatore-641 021, India
More informationThe number of simple modules of a cellular algebra
Science in China Ser. A Mathematics 2005 Vol.?? No. X: XXX XXX 1 The number of simple modules of a cellular algebra LI Weixia ( ) & XI Changchang ( ) School of Mathematical Sciences, Beijing Normal University,
More informationJordan α-centralizers in rings and some applications
Bol. Soc. Paran. Mat. (3s.) v. 26 1-2 (2008): 71 80. c SPM ISNN-00378712 Jordan α-centralizers in rings and some applications Shakir Ali and Claus Haetinger abstract: Let R be a ring, and α be an endomorphism
More informationMATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products.
MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products. Orthogonal projection Theorem 1 Let V be a subspace of R n. Then any vector x R n is uniquely represented
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationZero-Divisor Graph of Triangular Matrix Rings over Commutative Rings 1
International Journal of Algebra, Vol 5, 2011, no 6, 255-260 Zero-Divisor Graph of Triangular Matrix Rings over Commutative Rings 1 Li Bingjun 1,2 1 College of Mathematics Central South University Changsha,
More informationarxiv:math/ v1 [math.qa] 22 Nov 2005
Derivation Algebras of Centerless Perfect Lie Algebras Are Complete 1 (appeared in J. Algebra, 285 (2005), 508 515.) arxiv:math/0511550v1 [math.qa] 22 Nov 2005 Yucai Su, Linsheng Zhu Department of Mathematics,
More informationA Generalization of VNL-Rings and P P -Rings
Journal of Mathematical Research with Applications Mar, 2017, Vol 37, No 2, pp 199 208 DOI:103770/jissn:2095-2651201702008 Http://jmredluteducn A Generalization of VNL-Rings and P P -Rings Yueming XIANG
More informationJordan isomorphisms of triangular matrix algebras over a connected commutative ring
Linear Algebra and its Applications 312 (2000) 197 201 www.elsevier.com/locate/laa Jordan isomorphisms of triangular matrix algebras over a connected commutative ring K.I. Beidar a,,m.brešar b,1, M.A.
More informationLinear maps. Matthew Macauley. Department of Mathematical Sciences Clemson University Math 8530, Spring 2017
Linear maps Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Spring 2017 M. Macauley (Clemson) Linear maps Math 8530, Spring 2017
More informationLie Ideals and Generalized Derivations. in -Prime Rings - II
International Journal of Algebra, Vol. 6, 2012, no. 29, 1419 1429 Lie Ideals and Generalized Derivations in -Prime Rings - II M. S. Khan Department of Mathematics and Statistics Faculty of Science, Sultan
More informationLIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS
LIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS DIETRICH BURDE Abstract. We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify
More informationOn generalized -derivations in -rings
Palestine Journal of Mathematics Vol. 1 (2012), 32 37 Palestine Polytechnic University-PPU 2012 On generalized -derivations in -rings Shakir Ali Communicated by Tariq Rizvi 2000 Mathematics Subject Classification:
More informationSCHUR IDEALS AND HOMOMORPHISMS OF THE SEMIDEFINITE CONE
SCHUR IDEALS AND HOMOMORPHISMS OF THE SEMIDEFINITE CONE BABHRU JOSHI AND M. SEETHARAMA GOWDA Abstract. We consider the semidefinite cone K n consisting of all n n real symmetric positive semidefinite matrices.
More informationOn the Least Eigenvalue of Graphs with Cut Vertices
Journal of Mathematical Research & Exposition Nov., 010, Vol. 30, No. 6, pp. 951 956 DOI:10.3770/j.issn:1000-341X.010.06.001 Http://jmre.dlut.edu.cn On the Least Eigenvalue of Graphs with Cut Vertices
More informationHIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS
HIGHER CLASS GROUPS OF GENERALIZED EICHLER ORDERS XUEJUN GUO 1 ADEREMI KUKU 2 1 Department of Mathematics, Nanjing University Nanjing, Jiangsu 210093, The People s Republic of China guoxj@nju.edu.cn The
More informationComplete q-moment Convergence of Moving Average Processes under ϕ-mixing Assumption
Journal of Mathematical Research & Exposition Jul., 211, Vol.31, No.4, pp. 687 697 DOI:1.377/j.issn:1-341X.211.4.14 Http://jmre.dlut.edu.cn Complete q-moment Convergence of Moving Average Processes under
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationPermutation transformations of tensors with an application
DOI 10.1186/s40064-016-3720-1 RESEARCH Open Access Permutation transformations of tensors with an application Yao Tang Li *, Zheng Bo Li, Qi Long Liu and Qiong Liu *Correspondence: liyaotang@ynu.edu.cn
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F ) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationA note on restricted Lie supertriple systems
Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 10, Number 1 (2015), pp. 1 13 c Research India Publications http://www.ripublication.com/atam.htm A note on restricted Lie supertriple
More informationStrongly nil -clean rings
J. Algebra Comb. Discrete Appl. 4(2) 155 164 Received: 12 June 2015 Accepted: 20 February 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Strongly nil -clean rings Research Article
More informationDihedral groups of automorphisms of compact Riemann surfaces of genus two
Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article 36 2013 Dihedral groups of automorphisms of compact Riemann surfaces of genus two Qingje Yang yangqj@ruc.edu.com Dan Yang Follow
More informationTHE JORDAN-CHEVALLEY DECOMPOSITION
THE JORDAN-CHEVALLEY DECOMPOSITION JOO HEON YOO Abstract. This paper illustrates the Jordan-Chevalley decomposition through two related problems. Contents 1. Introduction 1 2. Linear Algebra Review 1 3.
More informationAsymptotic behavior for sums of non-identically distributed random variables
Appl. Math. J. Chinese Univ. 2019, 34(1: 45-54 Asymptotic behavior for sums of non-identically distributed random variables YU Chang-jun 1 CHENG Dong-ya 2,3 Abstract. For any given positive integer m,
More informationFixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc
Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 48, -9; http://www.math.u-szeged.hu/ejqtde/ Fixed points of the derivative and -th power of solutions of complex linear differential
More informationW P ZI rings and strong regularity
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 1 W P ZI rings and strong regularity Junchao Wei Received: 21.I.2013 / Revised: 12.VI.2013 / Accepted: 13.VI.2013 Abstract In this
More information4.1 Primary Decompositions
4.1 Primary Decompositions generalization of factorization of an integer as a product of prime powers. unique factorization of ideals in a large class of rings. In Z, a prime number p gives rise to a prime
More informationAbel rings and super-strongly clean rings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 2017, f. 2 Abel rings and super-strongly clean rings Yinchun Qu Junchao Wei Received: 11.IV.2013 / Last revision: 10.XII.2013 / Accepted: 12.XII.2013
More informationON THE MATRIX EQUATION XA AX = X P
ON THE MATRIX EQUATION XA AX = X P DIETRICH BURDE Abstract We study the matrix equation XA AX = X p in M n (K) for 1 < p < n It is shown that every matrix solution X is nilpotent and that the generalized
More informationIntegral Extensions. Chapter Integral Elements Definitions and Comments Lemma
Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients
More informationarxiv: v1 [math.ra] 28 Jan 2016
The Moore-Penrose inverse in rings with involution arxiv:1601.07685v1 [math.ra] 28 Jan 2016 Sanzhang Xu and Jianlong Chen Department of Mathematics, Southeast University, Nanjing 210096, China Abstract:
More informationarxiv: v1 [math.ra] 23 Feb 2018
JORDAN DERIVATIONS ON SEMIRINGS OF TRIANGULAR MATRICES arxiv:180208704v1 [mathra] 23 Feb 2018 Abstract Dimitrinka Vladeva University of forestry, bulklohridski 10, Sofia 1000, Bulgaria E-mail: d vladeva@abvbg
More informationPolynomials of small degree evaluated on matrices
Polynomials of small degree evaluated on matrices Zachary Mesyan January 1, 2013 Abstract A celebrated theorem of Shoda states that over any field K (of characteristic 0), every matrix with trace 0 can
More informationMULTIVARIATE BIRKHOFF-LAGRANGE INTERPOLATION SCHEMES AND CARTESIAN SETS OF NODES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXIII, 2(2004), pp. 217 221 217 MULTIVARIATE BIRKHOFF-LAGRANGE INTERPOLATION SCHEMES AND CARTESIAN SETS OF NODES N. CRAINIC Abstract. In this paper we study the relevance
More informationManoj K. Keshari 1 and Satya Mandal 2.
K 0 of hypersurfaces defined by x 2 1 +... + x 2 n = ±1 Manoj K. Keshari 1 and Satya Mandal 2 1 Department of Mathematics, IIT Mumbai, Mumbai - 400076, India 2 Department of Mathematics, University of
More informationBilateral truncated Jacobi s identity
Bilateral truncated Jacobi s identity Thomas Y He, Kathy Q Ji and Wenston JT Zang 3,3 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 30007, PR China Center for Applied Mathematics Tianjin
More informationCartan s Criteria. Math 649, Dan Barbasch. February 26
Cartan s Criteria Math 649, 2013 Dan Barbasch February 26 Cartan s Criteria REFERENCES: Humphreys, I.2 and I.3. Definition The Cartan-Killing form of a Lie algebra is the bilinear form B(x, y) := Tr(ad
More informationRANK AND PERIMETER PRESERVER OF RANK-1 MATRICES OVER MAX ALGEBRA
Discussiones Mathematicae General Algebra and Applications 23 (2003 ) 125 137 RANK AND PERIMETER PRESERVER OF RANK-1 MATRICES OVER MAX ALGEBRA Seok-Zun Song and Kyung-Tae Kang Department of Mathematics,
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationFormulas for the Drazin Inverse of Matrices over Skew Fields
Filomat 30:12 2016 3377 3388 DOI 102298/FIL1612377S Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://wwwpmfniacrs/filomat Formulas for the Drazin Inverse of
More informationarxiv:math/ v2 [math.ra] 14 Dec 2005
arxiv:math/0512273v2 [math.ra] 14 Dec 2005 AUTOMORPHISMS OF THE SEMIGROUP OF ENDOMORPHISMS OF FREE ASSOCIATIVE ALGEBRAS A. KANEL-BELOV, A. BERZINS AND R. LIPYANSKI Abstract. Let A = A(x 1,...,x n) be a
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationDefinitions and Properties of R N
Definitions and Properties of R N R N as a set As a set R n is simply the set of all ordered n-tuples (x 1,, x N ), called vectors. We usually denote the vector (x 1,, x N ), (y 1,, y N ), by x, y, or
More informationConjugacy classes of torsion in GL_n(Z)
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 32 2015 Conjugacy classes of torsion in GL_n(Z) Qingjie Yang Renmin University of China yangqj@ruceducn Follow this and additional
More informationGroup inverse for the block matrix with two identical subblocks over skew fields
Electronic Journal of Linear Algebra Volume 21 Volume 21 2010 Article 7 2010 Group inverse for the block matrix with two identical subblocks over skew fields Jiemei Zhao Changjiang Bu Follow this and additional
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationLie Weak Amenability of Triangular Banach Algebra
Journal of Mathematical Research with Applications Sept 217 Vol 37 No 5 pp 63 612 DOI:1377/jissn:295-265121751 Http://jmredluteducn ie Weak Amenaility of Triangular anach Algera in CHEN 1 Fangyan U 2 1
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More informationDERIVATIONS IN PRIME RINGS
proceedings of the american mathematical society Volume 84, Number 1, January 1982 DERIVATIONS IN PRIME RINGS B. FELZENSZWALB1 Abstract. Let R be a ring and d =0 a. derivation of R such that d(x") = 0,
More informationDICKSON POLYNOMIALS OVER FINITE FIELDS. n n i. i ( a) i x n 2i. y, a = yn+1 a n+1 /y n+1
DICKSON POLYNOMIALS OVER FINITE FIELDS QIANG WANG AND JOSEPH L. YUCAS Abstract. In this paper we introduce the notion of Dickson polynomials of the k + 1)-th kind over finite fields F p m and study basic
More informationLecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector
More informationBERNARD RUSSO University of California, Irvine. Report on joint work with. Antonio M. Peralta Universidad de Granada, Spain
TERNARY WEAKLY AMENABLE C*-ALGEBRAS BERNARD RUSSO University of California, Irvine Report on joint work with Antonio M. Peralta Universidad de Granada, Spain GPOTS XXXI ARIZONA STATE UNIVERSITY MAY 18,
More informationGeneralized (α, β)-derivations on Jordan ideals in -prime rings
Rend. Circ. Mat. Palermo (2014) 63:11 17 DOI 10.1007/s12215-013-0138-2 Generalized (α, β)-derivations on Jordan ideals in -prime rings Öznur Gölbaşi Özlem Kizilgöz Received: 20 May 2013 / Accepted: 7 October
More informationON THE SUBGROUPS OF TORSION-FREE GROUPS WHICH ARE SUBRINGS IN EVERY RING
italian journal of pure and applied mathematics n. 31 2013 (63 76) 63 ON THE SUBGROUPS OF TORSION-FREE GROUPS WHICH ARE SUBRINGS IN EVERY RING A.M. Aghdam Department Of Mathematics University of Tabriz
More information204 H. Almutari, A.G. Ahmad
International Journal of Pure and Applied Mathematics Volume 113 No. 2 2017, 203-217 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v113i2.2
More informationGreen s Function for an n-point Right Focal Boundary Value Problem
Green s Function for an n-point Right Focal Boundary Value Problem Doug Anderson Department of Mathematics and Computer Science, Concordia College Moorhead, MN 56562, USA Abstract We determine sufficient
More informationu n 2 4 u n 36 u n 1, n 1.
Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then
More informationOn EP elements, normal elements and partial isometries in rings with involution
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 39 2012 On EP elements, normal elements and partial isometries in rings with involution Weixing Chen wxchen5888@163.com Follow this
More informationThe Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices
Filomat 28:1 (2014, 65 71 DOI 10.2298/FIL1401065H Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Residual Spectrum and the
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Noncommutative Contact Algebras Hideki Omori Yoshiaki Maeda Naoya Miyazaki Akira Yoshioka
More information1.4 Solvable Lie algebras
1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)
More informationover a field F with char F 2: we define
Chapter 3 Involutions In this chapter, we define the standard involution (also called conjugation) on a quaternion algebra. In this way, we characterize division quaternion algebras as noncommutative division
More informationHIGHER-ORDER DIFFERENCES AND HIGHER-ORDER PARTIAL SUMS OF EULER S PARTITION FUNCTION
ISSN 2066-6594 Ann Acad Rom Sci Ser Math Appl Vol 10, No 1/2018 HIGHER-ORDER DIFFERENCES AND HIGHER-ORDER PARTIAL SUMS OF EULER S PARTITION FUNCTION Mircea Merca Dedicated to Professor Mihail Megan on
More informationLinear algebra. S. Richard
Linear algebra S. Richard Fall Semester 2014 and Spring Semester 2015 2 Contents Introduction 5 0.1 Motivation.................................. 5 1 Geometric setting 7 1.1 The Euclidean space R n..........................
More informationGENERATING NON-NOETHERIAN MODULES CONSTRUCTIVELY
GENERATING NON-NOETHERIAN MODULES CONSTRUCTIVELY THIERRY COQUAND, HENRI LOMBARDI, CLAUDE QUITTÉ Abstract. In [6], Heitmann gives a proof of a Basic Element Theorem, which has as corollaries some versions
More informationAn Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation
Commun. Theor. Phys. (Beijing, China) 50 (008) pp. 309 314 c Chinese Physical Society Vol. 50, No., August 15, 008 An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson
More informationMATH 431: FIRST MIDTERM. Thursday, October 3, 2013.
MATH 431: FIRST MIDTERM Thursday, October 3, 213. (1) An inner product on the space of matrices. Let V be the vector space of 2 2 real matrices (that is, the algebra Mat 2 (R), but without the mulitiplicative
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationTHEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)
4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M
More informationProlongation structure for nonlinear integrable couplings of a KdV soliton hierarchy
Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy Yu Fa-Jun School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang 110034, China Received
More informationGeneralized core inverses of matrices
Generalized core inverses of matrices Sanzhang Xu, Jianlong Chen, Julio Benítez and Dingguo Wang arxiv:179.4476v1 [math.ra 13 Sep 217 Abstract: In this paper, we introduce two new generalized inverses
More informationACI-matrices all of whose completions have the same rank
ACI-matrices all of whose completions have the same rank Zejun Huang, Xingzhi Zhan Department of Mathematics East China Normal University Shanghai 200241, China Abstract We characterize the ACI-matrices
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationSTRONGLY J-CLEAN SKEW TRIANGULAR MATRIX RINGS *
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul...,..., f... DOI: 10.1515/aicu-2015-0008 STRONGLY J-CLEAN SKEW TRIANGULAR MATRIX RINGS * BY YOSUM KURTULMAZ Abstract.
More informationMath 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.
Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses
More informationMath 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March 22, 2018
1 Linear Systems Math 5630: Iterative Methods for Systems of Equations Hung Phan, UMass Lowell March, 018 Consider the system 4x y + z = 7 4x 8y + z = 1 x + y + 5z = 15. We then obtain x = 1 4 (7 + y z)
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationMODEL THEORY OF PARTIAL DIFFERENTIAL FIELDS: FROM COMMUTING TO NONCOMMUTING DERIVATIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MODEL THEORY OF PARTIAL DIFFERENTIAL FIELDS: FROM COMMUTING TO NONCOMMUTING DERIVATIONS MICHAEL
More informationGeneralized Multiplicative Derivations in Near-Rings
Generalized Multiplicative Derivations in Near-Rings Mohammad Aslam Siddeeque Department of Mathematics Aligarh Muslim University Aligarh -222(India) E-mail : aslamsiddeeque@gmail.com Abstract: In the
More informationNumerical Analysis: Interpolation Part 1
Numerical Analysis: Interpolation Part 1 Computer Science, Ben-Gurion University (slides based mostly on Prof. Ben-Shahar s notes) 2018/2019, Fall Semester BGU CS Interpolation (ver. 1.00) AY 2018/2019,
More informationAnn. Funct. Anal. 5 (2014), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann Funct Anal 5 (2014), no 2, 127 137 A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL:wwwemisde/journals/AFA/ THE ROOTS AND LINKS IN A CLASS OF M-MATRICES XIAO-DONG ZHANG This paper
More informationOn Projective Planes
C-UPPSATS 2002:02 TFM, Mid Sweden University 851 70 Sundsvall Tel: 060-14 86 00 On Projective Planes 1 2 7 4 3 6 5 The Fano plane, the smallest projective plane. By Johan Kåhrström ii iii Abstract It was
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationNew lower bounds for hypergraph Ramsey numbers
New lower bounds for hypergraph Ramsey numbers Dhruv Mubayi Andrew Suk Abstract The Ramsey number r k (s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1,..., N}, there
More informationHouston Journal of Mathematics. c 2012 University of Houston Volume 38, No. 1, Communicated by Kenneth R. Davidson
Houston Journal of Mathematics c 2012 University of Houston Volume 38, No. 1, 2012 JORDAN HIGHER DERIVATIONS ON SOME OPERATOR ALGEBRAS ZHANKUI XIAO AND FENG WEI Communicated by Kenneth R. Davidson Abstract.
More information