Generalized Boolean and Boolean-Like Rings
|
|
- Lora Casey
- 5 years ago
- Views:
Transcription
1 International Journal of Algebra, Vol. 7, 2013, no. 9, HIKARI Ltd, Generalized Boolean and Boolean-Like Rings Hazar Abu Khuzam Department of Mathematics American University of Beirut Beirut, Lebanon Adil Yaqub Department of Mathematics University of California Santa Barbara, CA , USA Copyright 2013 Hazar Abu Khuzam and Adil Yaqub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A generalized Boolean ring is a ring R such that, for all x, y in R\(N C), x n y xy n (N C), where n is a fixed even integer, and N, C are the set of nilpotents and center of R, respectively. A Boolean-like ring is a ring R such that, for all x,y R\(N C), x n y xy n, where again n is a fixed even integer. The commutativity behavior of these rings is considered. In particular, it is proved that a Boolean-like ring with identity is commutative.
2 430 Hazar Abu Khuzam and Adil Yaqub Mathematics Subject Classification: 16U80, 16D70 Keywords: Generalized Boolean ring, Boolean-like ring, Jacobson radical, commutator ideal, weakly periodic-like ring Throughout R is a ring, N is the set of nilpotents, J is the Jacobson radical, and C is the center of R. As usual, [x,y] denotes the commutator xy yx. Definition 1. Let n be a fixed positive even integer. A ring R is called a generalized Boolean ring if (1) x n y xy n (N C) for all x, y in R\(N C), (n even). R is called a Boolean - like ring if (2) x n y xy n = 0 for all x, y R\(N C), (n even). In preparation for the proofs of the main theorems, we state the following lemmas. Lemma 1. ([1]) Suppose R is a ring in which each element x is central or potent in the sense that x k = x for some k > 1. Then R is commutative. Lemma 2. Suppose R is a ring and x R. Suppose that (3) x m x m+1 f (x) N, m a positive integer and f ( λ ) Z [ λ ]. Then x x 2 f(x) N. Proof. Note that (x x 2 f(x)) m = (x x 2 f(x) ) x m-1 g(x), for some g ( λ ) Z [ λ ]. = ( x m x m+1 f (x) ) g(x) N, and hence x x 2 f(x) N, proving the lemma. Lemma 3. Suppose R is a generalized Boolean ring. Then, (4) For all x R, either x C or 2x N.
3 Generalized Boolean and Boolean-like rings 431 Proof. By contradiction. Suppose that x R, x C and 2x N. Then x C and x N, and hence by (1), x n ( x) x( x) n N, n even. Hence 2x n+1 N, which implies that 2x N, contradiction, which proves the lemma. Lemma 4. Suppose R is a generalized Boolean ring, and suppose all idempotents are central. Then, N J, (J is the Jacobson radical of R). Proof. Let a N, x R. If ax N, then ax is right quasi regular (r.q.r.). Also, since ax C implies ax N, it follows that ax C implies ax is r.q.r. A similar argument shows that (ax) 2 N implies ax is r.q.r. Moreover, (ax) 2 C implies ((ax) 2 ) k = (ax) 2 (ax) 2 (ax) 2 = a k y for some y in R, k any positive integer. Since a N, a k = 0 for some k, and thus ((ax) 2 ) k = 0, which implies ax N. So ax is r.q.r. again. The only case left to consider is: ax (N C) and (ax) 2 (N C). By (1), (ax) n (ax) 2 (ax) ((ax) 2 ) n N and thus (ax) n+2 (ax) 2n+1 N. Since n+2 2n+1, we conclude that (ax) q = (ax) q+1 g(ax), g ( λ ) Z [ λ ], and hence (ax) q = (ax) q (ax g(ax)) =. = (ax) q (ax g(ax)) q. Therefore, (5) (ax) q = (ax) q e, e 2 =e ar, e = (ax g(ax)) q. Hence, for some r R, e = e.e = e(ar) = aer ( since e is central). By re-iterating, we see that e = aer = a 2 er 2 = = a k er k for all positive integer k. Since a N, a k = 0 for some k, and thus e = 0, which implies by (5) that (ax) q =0. Thus, ax is r.q.r. This shows that ax is r.q.r. for all a N, x R, and hence a J, which proves the lemma. We are in a position to prove the main theorems Theorem 1. Any subring and any homomorphic image of a generalized Boolean ring is also a generalized Boolean ring. This follows readily from Definition 1.
4 432 Hazar Abu Khuzam and Adil Yaqub Theorem 2. If R is a generalized Boolean ring with central nilpotents, then R is commutative. Proof. Suppose x R, x C. We now distinguish two cases. Case 1. x 2 C. Since N C, (1) implies that (6) x n y x y n N for all x, y R\C. Also, since x C and x 2 C, it follows by (6) that x n (x 2 ) x(x 2 ) n N and hence x n+2 x 2n+1 N, which implies by Lemma 2 that x x n N C. Thus, x x n C for all x in R, and hence by a well known theorem of Herstein [2], R is commutative. Case 2. x 2 C. In this case, x x 2 C (since x C), and hence by (6) x n (x x 2 ) x( x x 2 ) n N, which implies that x n+1 x n+2 x(x n nx n+1 +x n+2 g(x)) N, for some g ( λ ) Z [ λ ]. Therefore, (7) (n 1)x n+2 x n+3 g(x) N. Since x C, 2x N (by Lemma 3), and hence nx N ( since n is even), which implies that (8) nx n+2 N. Thus, (n-1)x n+2 -x n+3 g(x) and nx n+2 are two commuting nilpotents, and hence their difference is in N; that is, x n+2 x n+3 g(x) N. Therefore, by the proof of Lemma 2, x x 2 g(x) N, which implies that x x 2 f(x) N, where f ( λ ) = g ( λ ) Z [ λ ]. Thus x x 2 f(x) C ( since N C ) for all x R, which implies that R is commutative, by [2]. This proves the theorem. Theorem 3. Suppose R is a generalized Boolean ring with identity. Then all idempotents of R are central. Proof. Let e 2 =e R, x R, and let a = xe exe, suppose a 0. Since a N, 1+ a N. Moreover, 1+a C, since 1+ a C implies that a C, and hence ae = ea; that is, a = 0, contradiction. Thus (1+ a) (N C). Moreover, a 0, e (N C). So, by (1), (1+a) n e (1+a) e n C, which implies that nae ae C; which implies (n 1)a = 0 since ae=a and ea=0. Moreover, since a 0, e 0, and hence -e 0, which implies that -e (N C), and e (N C). Therefore, by (1),
5 Generalized Boolean and Boolean-like rings 433 (-e) n e-(-e)e n C, and hence, since n is even, 2e C. Thus, a(2e) = (2e)a, which implies that 2a = 0. The net result is: (9) (n 1) a = 0 and 2a = 0. Since n is even, (9) implies that a=0, contradiction. Hence, a = 0, and thus xe = exe. Similarly, ex = exe, and the Theorem is proved. Theorem 4. Suppose R is a generalized Boolean ring with identity. Then N is an ideal and R/N is commutative (and thus the commutator ideal of R is nil). Proof. By Theorem 3, all idempotents of R are central, and hence by Lemma 4, N J, where J is the Jacobson radical of R. We claim that (10) J N C. Suppose not. Let j J, j (N C). Then, 1+j (N C), and hence by (1), (1+j) n j (1+j) j n N, which implies that j j 2 f(j) N, for some f ( λ ) Z [ λ ]. Therefore, j(1 j f(j)) N, and, moreover, 1 j f(j) is a unit in R. Thus, 1 1, and hence j N, contradiction. This contradiction proves (10). Hence (11) N J N C. We claim that (12) N is an ideal of R. To prove this, suppose a N, x R. Then, by (11), a J, x R and hence ax J N C (by (11)). Hence ax N or ax C (which also implies that ax N), which proves that ax N. Similarly xa N. Next, suppose a N, b N, Then a J, b J (by (11)), and hence a b J N C (by (11)). So, a b N or a b C. If a b C, then a commutes with b, and hence a b N again, which proves (12). Let S = R / N. Then, by (1), (13) x n y = xy n for all noncentral elements of S. Moreover, S has an identity. Now suppose x is a noncentral element of S. Then, 1+x is also a noncentral element of S, and hence by (13), x n (1+x) = x(1+x) n. Therefore, for all noncentral elements x in S, x x 2 f(x) = 0 for some f ( λ ) Z [ λ ], which implies x x 2 f(x) is in the center of S, for all elements x of S. Hence, by Herstein s theorem [2], S is commutative; that is, R/N is commutative, which proves the theorem. Theorem 5. Suppose R is a generalized Boolean ring with identity, and suppose (14) N J is commutative.
6 434 Hazar Abu Khuzam and Adil Yaqub Then R is commutative. Proof. By Theorem 4, N is an ideal, and hence N J, which implies by (14), that (15) N is a commutative ideal of R. We claim (16) [a, x]x = 0 for all a N, x R. The proof is by contradiction. Suppose that for some a N, x R, [a, x] x 0, which implies x N (by (15)) and, of course, x C. So x (N C). Moreover, 1 1+ a C (since [a, x] x 0) and 1 + a N, which implies that 1+a (N C). Hence, by (1), (17) (1+a) n x (1+a) x n N C. Now, since N is an ideal (see (15)) and a N, (17) implies that (18) x x n N. Moreover, since N is a commutative ideal (see (15)), N 2 C. Also, (17) clearly implies that (19) [(1+a) n x (1 +a) x n, x] = 0 (a N, N 2 C). Thus, by (19), (20) [nax ax n, x] = 0. Let x x n = a o. By (18), a o N, and thus (21) x n = x a o, (a o N). Combining (20) and (21), we get [nax a(x a o ), x ] = 0, and hence [nax ax, x ] = 0 since aa o N 2 C; that is, [(n 1)ax, x ] = 0. Therefore, (22) (n 1) [ax, x ] = 0, (a N, x R). Recall that, by Lemma 3, x C or 2x N. If x C, then clearly [ax, 2x] = 0. Moreover, if 2x N, then [ax, 2x] = 0, by (15) which implies (23) 2 [ax, x] = [ax, 2x] = 0 (a N, x R). Combining (22) and (23), we see that (n 1) [ax, x] = 0 and 2 [ax, x] = 0, and n is even. So (24) [a, x] x = [ax, x] = 0, contradiction. This contradiction proves (16).
7 Generalized Boolean and Boolean-like rings 435 To complete the proof, since (16) is true for all x in R, we may replace x by x+1 in the above argument to obtain (see (24)) (25) [a, x+1] (x+1) = 0. In view of (24) and (25), we see that [a, x] = 0 for all a N, x R, which proves that (26) N C The theorem now follows from Theorem 2 and (26). In order to deal with the case where R does not have an identity in Theorem 5, we introduce the following Definition 2. A ring R is called weakly periodic-like if every element x in R\C is of the form (27) x = a + b, a N, b potent in the sense that b k = b for some k>1, x C. Theorem 6. Suppose R is a generalized Boolean ring, not necessarily with identity, and suppose, further, R is a weakly periodic- like ring and all the idempotents of R are central. Then N is an ideal and R/N is commutative (and thus the commutator ideal of R is nil). Proof. By Lemma 4, N C. We now prove that (28) J N C. Suppose not, let j J, j N, j C. Then, by (27), (29) j = a +b, a N, b k = b for some k>1, (j C). By (29), we see that (30) j a = ( j a) k, and hence = for all positive integers m. Since a N, = 0 for some m 1, and hence by (30), j a J; that is b J (see (29)). Hence, by (29), b k 1 is an idempotent element in J, and thus b k 1 = 0. So, b = b k = 0, and hence by (29), j = a N, contradiction. This proves (28). A combination of N J and (28) yields (31) N J NU C In the proof of Theorem 4, it was shown that in any ring R, (31) implies that (32) N is an ideal. Next we show that (33) for any x R, either x C or x - x k N for some k>1. To see this, note that by Definition 2, (34) x = a + b, a N, b k = b for some k>1, x R\C,
8 436 Hazar Abu Khuzam and Adil Yaqub and hence (35) x a = ( x a) k, k>1, (a N, x R\C). Combining (32) and (35), we obtain (33). Therefore, by (32) and (33), we see that every element of R/N is central or potent (satisfying x k = x, x R/N, k>1), and hence by a Theorem of Bell (Lemma 1), R/N is commutative, which proves the theorem. Theorem 7. Suppose R is a generalized Boolean ring and, moreover, R is weakly periodic-like. Suppose, further that all the idempotents of R are central and N J is commutative. Then R is commutative. Proof. By Lemma 4, N J, and hence N = N J is commutative. Thus, (36) N is commutative. Next we show that (37) All potent elements of R are central. Suppose that b is a potent nonzero element in R, and suppose b k = b, k>1. Let e = b k 1. Then, e is a nonzero idempotent of R and, moreover, e is central (by hypothesis). Hence, er is a ring with identity e. It is readily verified that er satisfies all the hypotheses imposed on R in Theorem 5. (To verify that (14) holds in er, keep in mind, that J(eR) = ej(r) J(R)). Thus, by Theorem 5, (38) er is commutative. Let y N. Then, by (38), e [b, y] = [eb, ey] = 0. Recalling that e 2 = e = b k-1 C and b k = b, we see that 0 = e [b, y] = b k-1 [b, y] = b k y b k 1 yb = b k y yb k = by yb, and hence by = yb for all y R, which proves (37). To complete the proof, suppose x R\C, y R\C. Then, by (27), (39) x = a + b, a N, b potent, and also y = a' + b', a' N, b' potent, and hence [x, y] = [a + b, a' + b']=[a, a'], by (37), and [a, a'] = 0, by (36). Hence [x, y] = 0, and thus R is commutative, which proves the Theorem. We now focus our attention on Boolean- like rings. Theorem 8. A Boolean-like ring with identity is commutative. Proof. First, we prove that
9 Generalized Boolean and Boolean-like rings 437 (40) for all a N and all units u R, [a, u] = 0. To prove this, suppose that (since a N) (41) [a σ, u] = 0 for all σ σ o, σ o minimal. We claim that (42) σ o = 1. Suppose not. Then, by the minimality of σ o in (41), 1 + a σo-1 C, and of course 1 + a σo-1 N (since a N). If u C, then (40) is trivially satisfied. So we may assume that u C, and of course u N. Hence, by (2), (1 + a σo-1 ) n u = ( 1 + a σo-1 ) u n, n even, and thus (since 1 + a σo-1 is a unit) ( 1 + a σo-1 ) n-1 = u n 1 which implies that (43) [( 1 + a σo-1 ) n-1, u] =0. By (43) and (41), we see that (44) (n 1) [a σo-1, u] =0, n even. Since u C, 2u N ( by Lemma 3), and hence (2u)u -1 N; that is, 2 N, which implies that R is of characteristic 2 k for some k 1, and thus (45) 2 k [a σo-1, u] =0. Since n is even, (44) and (45) imply that [a σo-1, u] =0, which contradicts the minimality of σ o in (41). This contradiction proves that σ o = 1, and hence by (41), [a, u] = 0, which proves (40). Next, we prove that (46) N is commutative. Let a N, b N. Then 1 + b is a unit in R, and hence by (40), [a, 1+b] =0, which implies [a, b] = 0, and (46) is proved. Hence by Theorem 5 and (46), R is commutative. We conclude with the following: Remark 1. Theorem 5 and 7 are not true if the hypothesis n is even is deleted in Definition 1, as can be seen by taking R = a b c 0 0 a 2 0 a 0 : a,b,c GF(4) ; n = 7.
10 438 Hazar Abu Khuzam and Adil Yaqub Indeed all the hypotheses of Theorem 5 and 7 hold except that n isn t even. But R is not commutative. Remark 2. Theorem 8 is not true if R does not have an identity, as can be seen by taking R = ,,, : 0,1 GF(2), n = Related work appears in [3]. References [1] H.E. Bell, A near-commutativity property for rings, Result. Math. 42 (2002), [2] I.N. Herstein, A gemeralization of a theorem of Jacobson III, Amer. J. Math. 75 (1953), [3] A. Yaqub, A generalization of Boolean rings, Intern. J. Algebra, 1 (2007), Received: August 5, 2012
A Generalization of p-rings
International Journal of Algebra, Vol. 9, 2015, no. 8, 395-401 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5848 A Generalization of p-rings Adil Yaqub Department of Mathematics University
More informationA Generalization of Boolean Rings
A Generalization of Boolean Rings Adil Yaqub Abstract: A Boolean ring satisfies the identity x 2 = x which, of course, implies the identity x 2 y xy 2 = 0. With this as motivation, we define a subboolean
More informationWEAKLY PERIODIC AND SUBWEAKLY PERIODIC RINGS
IJMMS 2003:33, 2097 2107 PII. S0161171203210589 http://ijmms.hindawi.com Hindawi Publishing Corp. WEAKLY PERIODIC AND SUBWEAKLY PERIODIC RINGS AMBER ROSIN and ADIL YAQUB Received 30 October 2002 Our objective
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 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 informationRINGS IN WHICH EVERY ZERO DIVISOR IS THE SUM OR DIFFERENCE OF A NILPOTENT ELEMENT AND AN IDEMPOTENT
RINGS IN WHICH EVERY ZERO DIVISOR IS THE SUM OR DIFFERENCE OF A NILPOTENT ELEMENT AND AN IDEMPOTENT MARJAN SHEBANI ABDOLYOUSEFI and HUANYIN CHEN Communicated by Vasile Brînzănescu An element in a ring
More informationA New Characterization of Boolean Rings with Identity
Irish Math. Soc. Bulletin Number 76, Winter 2015, 55 60 ISSN 0791-5578 A New Characterization of Boolean Rings with Identity PETER DANCHEV Abstract. We define the class of nil-regular rings and show that
More informationON STRUCTURE AND COMMUTATIVITY OF NEAR - RINGS
Proyecciones Vol. 19, N o 2, pp. 113-124, August 2000 Universidad Católica del Norte Antofagasta - Chile ON STRUCTURE AND COMMUTATIVITY OF NEAR - RINGS H. A. S. ABUJABAL, M. A. OBAID and M. A. KHAN King
More informationOn Reflexive Rings with Involution
International Journal of Algebra, Vol. 12, 2018, no. 3, 115-132 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8412 On Reflexive Rings with Involution Usama A. Aburawash and Muna E. Abdulhafed
More informationCharacterization of Weakly Primary Ideals over Non-commutative Rings
International Mathematical Forum, Vol. 9, 2014, no. 34, 1659-1667 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.49155 Characterization of Weakly Primary Ideals over Non-commutative Rings
More informationDiameter of the Zero Divisor Graph of Semiring of Matrices over Boolean Semiring
International Mathematical Forum, Vol. 9, 2014, no. 29, 1369-1375 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47131 Diameter of the Zero Divisor Graph of Semiring of Matrices over
More informationUnit Group of Z 2 D 10
International Journal of Algebra, Vol. 9, 2015, no. 4, 179-183 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5420 Unit Group of Z 2 D 10 Parvesh Kumari Department of Mathematics Indian
More information1.5 The Nil and Jacobson Radicals
1.5 The Nil and Jacobson Radicals The idea of a radical of a ring A is an ideal I comprising some nasty piece of A such that A/I is well-behaved or tractable. Two types considered here are the nil and
More informationEP elements and Strongly Regular Rings
Filomat 32:1 (2018), 117 125 https://doi.org/10.2298/fil1801117y Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat EP elements and
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 Generalized Derivations and Commutativity. of Prime Rings with Involution
International Journal of Algebra, Vol. 11, 2017, no. 6, 291-300 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7839 On Generalized Derivations and Commutativity of Prime Rings with Involution
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 informationInternational Journal of Algebra, Vol. 7, 2013, no. 3, HIKARI Ltd, On KUS-Algebras. and Areej T.
International Journal of Algebra, Vol. 7, 2013, no. 3, 131-144 HIKARI Ltd, www.m-hikari.com On KUS-Algebras Samy M. Mostafa a, Mokhtar A. Abdel Naby a, Fayza Abdel Halim b and Areej T. Hameed b a Department
More informationOn Permutation Polynomials over Local Finite Commutative Rings
International Journal of Algebra, Vol. 12, 2018, no. 7, 285-295 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8935 On Permutation Polynomials over Local Finite Commutative Rings Javier
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationSome Polynomial Identities that Imply Commutativity of Rings
International Journal of Algebra, Vol. 4, 2010, no. 27, 1307-1316 Some Polynomial Identities that Imply Commutativity of Rings M. S. Khan Department of Mathematics and Statistics College of Science, P.O.
More informationWeakly-Abel rings and strongly regular rings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 2016, f. 2, vol. 1 Weakly-Abel rings and strongly regular rings Jianhua Chen Junchao Wei Received: 1.XII.2013 / Accepted: 15.V.2014 Abstract In
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationON STRONGLY REGULAR RINGS AND GENERALIZATIONS OF V -RINGS. Tikaram Subedi and Ardeline Mary Buhphang
International Electronic Journal of Algebra Volume 14 (2013) 10-18 ON STRONGLY REGULAR RINGS AND GENERALIZATIONS OF V -RINGS Tikaram Subedi and Ardeline Mary Buhphang Received: 3 April 2012; Revised: 4
More informationReview Article Some Notes on Semiabelian Rings
International Mathematics and Mathematical Sciences Volume 2011, Article ID 154636, 10 pages doi:10.1155/2011/154636 Review Article Some Notes on Semiabelian Rings Junchao Wei and Nanjie Li School of Mathematics,
More informationInternational Mathematical Forum, Vol. 9, 2014, no. 36, HIKARI Ltd,
International Mathematical Forum, Vol. 9, 2014, no. 36, 1751-1756 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.411187 Generalized Filters S. Palaniammal Department of Mathematics Thiruvalluvar
More informationPrime and Semiprime Bi-ideals in Ordered Semigroups
International Journal of Algebra, Vol. 7, 2013, no. 17, 839-845 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.310105 Prime and Semiprime Bi-ideals in Ordered Semigroups R. Saritha Department
More informationISOMORPHIC POLYNOMIAL RINGS
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 27, No. 2, February 1971 ISOMORPHIC POLYNOMIAL RINGS D. B. COLEMAN AND E. E. ENOCHS Abstract. A ring is called invariant if whenever B is a ring
More informationr-ideals of Commutative Semigroups
International Journal of Algebra, Vol. 10, 2016, no. 11, 525-533 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.61276 r-ideals of Commutative Semigroups Muhammet Ali Erbay Department of
More informationWhen is the Ring of 2x2 Matrices over a Ring Galois?
International Journal of Algebra, Vol. 7, 2013, no. 9, 439-444 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.3445 When is the Ring of 2x2 Matrices over a Ring Galois? Audrey Nelson Department
More informationON NIL SEMI CLEAN RINGS *
Jordan Journal of Mathematics and Statistics (JJMS) 2 (2), 2009, pp. 95-103 ON NIL SEMI CLEAN RINGS * MOHAMED KHEIR AHMAD OMAR AL-MALLAH ABSTRACT: In this paper, the notions of semi-idempotent elements
More informationOn the Power of Standard Polynomial to M a,b (E)
International Journal of Algebra, Vol. 10, 2016, no. 4, 171-177 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6214 On the Power of Standard Polynomial to M a,b (E) Fernanda G. de Paula
More informationContra θ-c-continuous Functions
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 1, 43-50 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.714 Contra θ-c-continuous Functions C. W. Baker
More informationWritten Homework # 4 Solution
Math 516 Fall 2006 Radford Written Homework # 4 Solution 12/10/06 You may use results form the book in Chapters 1 6 of the text, from notes found on our course web page, and results of the previous homework.
More informationPre-Hilbert Absolute-Valued Algebras Satisfying (x, x 2, x) = (x 2, y, x 2 ) = 0
International Journal of Algebra, Vol. 10, 2016, no. 9, 437-450 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6743 Pre-Hilbert Absolute-Valued Algebras Satisfying (x, x 2, x = (x 2,
More informationA Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings
Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive
More informationChan Huh, Sung Hee Jang, Chol On Kim, and Yang Lee
Bull. Korean Math. Soc. 39 (2002), No. 3, pp. 411 422 RINGS WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED Chan Huh, Sung Hee Jang, Chol On Kim, and Yang Lee Abstract. In this note we are concerned with
More informationProve proposition 68. It states: Let R be a ring. We have the following
Theorem HW7.1. properties: Prove proposition 68. It states: Let R be a ring. We have the following 1. The ring R only has one additive identity. That is, if 0 R with 0 +b = b+0 = b for every b R, then
More informationRemarks on Fuglede-Putnam Theorem for Normal Operators Modulo the Hilbert-Schmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 1389-1396 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on Fuglede-Putnam Theorem for Normal Operators Modulo the
More informationThe Endomorphism Ring of a Galois Azumaya Extension
International Journal of Algebra, Vol. 7, 2013, no. 11, 527-532 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.29110 The Endomorphism Ring of a Galois Azumaya Extension Xiaolong Jiang
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 informationOn quasi-reduced rings
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 1 (2016), pp. 927 935 Research India Publications http://www.ripublication.com/gjpam.htm On quasi-reduced rings Sang Jo
More informationSome Properties of D-sets of a Group 1
International Mathematical Forum, Vol. 9, 2014, no. 21, 1035-1040 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.45104 Some Properties of D-sets of a Group 1 Joris N. Buloron, Cristopher
More informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is
More informationDirect Product of BF-Algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 125-132 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.614 Direct Product of BF-Algebras Randy C. Teves and Joemar C. Endam Department
More informationHong Kee Kim, Nam Kyun Kim, and Yang Lee
J. Korean Math. Soc. 42 (2005), No. 3, pp. 457 470 WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL Hong Kee Kim, Nam Kyun Kim, and Yang Lee Abstract. Yu showed that every right (left) primitive factor ring
More informationRight Derivations on Semirings
International Mathematical Forum, Vol. 8, 2013, no. 32, 1569-1576 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2013.38150 Right Derivations on Semirings S. P. Nirmala Devi Department of
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb
ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal
More informationOn Strongly Regular Rings and Generalizations of Semicommutative Rings
International Mathematical Forum, Vol. 7, 2012, no. 16, 777-790 On Strongly Regular Rings and Generalizations of Semicommutative Rings Tikaram Subedi Department of Mathematics North Eastern Hill University,
More informationComplete and Fuzzy Complete d s -Filter
International Journal of Mathematical Analysis Vol. 11, 2017, no. 14, 657-665 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.7684 Complete and Fuzzy Complete d s -Filter Habeeb Kareem
More informationOn J(R) of the Semilocal Rings
International Journal of Algebra, Vol. 11, 2017, no. 7, 311-320 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.61169 On J(R) of the Semilocal Rings Giovanni Di Gregorio Dipartimento di
More informationOn Annihilator Small Intersection Graph
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 7, 283-289 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7931 On Annihilator Small Intersection Graph Mehdi
More informationWeakly Semicommutative Rings and Strongly Regular Rings
KYUNGPOOK Math. J. 54(2014), 65-72 http://dx.doi.org/10.5666/kmj.2014.54.1.65 Weakly Semicommutative Rings and Strongly Regular Rings Long Wang School of Mathematics, Yangzhou University, Yangzhou, 225002,
More informationRestrained Independent 2-Domination in the Join and Corona of Graphs
Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs
More informationTripotents: a class of strongly clean elements in rings
DOI: 0.2478/auom-208-0003 An. Şt. Univ. Ovidius Constanţa Vol. 26(),208, 69 80 Tripotents: a class of strongly clean elements in rings Grigore Călugăreanu Abstract Periodic elements in a ring generate
More informationSubring of a SCS-Ring
International Journal of Algebra, Vol. 7, 2013, no. 18, 867-871 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.3986 Subring of a SCS-Ring Ishagh ould EBBATT, Sidy Demba TOURE, Abdoulaye
More informationOn Boolean Like Ring Extension of a Group
International Journal of Algebra, Vol. 8, 2014, no. 3, 121-128 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.312136 On Boolean Like Ring Extension of a Group Dawit Chernet and K. Venkateswarlu
More informationThe Automorphisms of a Lie algebra
Applied Mathematical Sciences Vol. 9 25 no. 3 2-27 HIKARI Ltd www.m-hikari.com http://dx.doi.org/.2988/ams.25.4895 The Automorphisms of a Lie algebra WonSok Yoo Department of Applied Mathematics Kumoh
More informationPrime Hyperideal in Multiplicative Ternary Hyperrings
International Journal of Algebra, Vol. 10, 2016, no. 5, 207-219 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6320 Prime Hyperideal in Multiplicative Ternary Hyperrings Md. Salim Department
More informationMappings of the Direct Product of B-algebras
International Journal of Algebra, Vol. 10, 2016, no. 3, 133-140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.615 Mappings of the Direct Product of B-algebras Jacel Angeline V. Lingcong
More informationOn (m,n)-ideals in LA-Semigroups
Applied Mathematical Sciences, Vol. 7, 2013, no. 44, 2187-2191 HIKARI Ltd, www.m-hikari.com On (m,n)-ideals in LA-Semigroups Muhammad Akram University of Gujrat Gujrat, Pakistan makram 69@yahoo.com Naveed
More informationSecure Weakly Connected Domination in the Join of Graphs
International Journal of Mathematical Analysis Vol. 9, 2015, no. 14, 697-702 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.519 Secure Weakly Connected Domination in the Join of Graphs
More informationCross Connection of Boolean Lattice
International Journal of Algebra, Vol. 11, 2017, no. 4, 171-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7419 Cross Connection of Boolean Lattice P. G. Romeo P. R. Sreejamol Dept.
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 informationSecure Weakly Convex Domination in Graphs
Applied Mathematical Sciences, Vol 9, 2015, no 3, 143-147 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams2015411992 Secure Weakly Convex Domination in Graphs Rene E Leonida Mathematics Department
More informationDetection Whether a Monoid of the Form N n / M is Affine or Not
International Journal of Algebra Vol 10 2016 no 7 313-325 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/1012988/ija20166637 Detection Whether a Monoid of the Form N n / M is Affine or Not Belgin Özer and Ece
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 informationEXERCISES. = {1, 4}, and. The zero coset is J. Thus, by (***), to say that J 4- a iu not zero, is to
19 CHAPTER NINETEEN Whenever J is a prime ideal of a commutative ring with unity A, the quotient ring A/J is an integral domain. (The details are left as an exercise.) An ideal of a ring is called proper
More informationOn the Computation of the Adjoint Ideal of Curves with Ordinary Singularities
Applied Mathematical Sciences Vol. 8, 2014, no. 136, 6805-6812 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49697 On the Computation of the Adjoint Ideal of Curves with Ordinary Singularities
More informationInternational Journal of Algebra, Vol. 4, 2010, no. 2, S. Uma
International Journal of Algebra, Vol. 4, 2010, no. 2, 71-79 α 1, α 2 Near-Rings S. Uma Department of Mathematics Kumaraguru College of Technology Coimbatore, India psumapadma@yahoo.co.in R. Balakrishnan
More informationRINGS HAVING ZERO-DIVISOR GRAPHS OF SMALL DIAMETER OR LARGE GIRTH. S.B. Mulay
Bull. Austral. Math. Soc. Vol. 72 (2005) [481 490] 13a99, 05c99 RINGS HAVING ZERO-DIVISOR GRAPHS OF SMALL DIAMETER OR LARGE GIRTH S.B. Mulay Let R be a commutative ring possessing (non-zero) zero-divisors.
More informationAlgebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.
More informationRestrained Weakly Connected Independent Domination in the Corona and Composition of Graphs
Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 973-978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121046 Restrained Weakly Connected Independent Domination in the Corona and
More informationGeneralization of the Banach Fixed Point Theorem for Mappings in (R, ϕ)-spaces
International Mathematical Forum, Vol. 10, 2015, no. 12, 579-585 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.5861 Generalization of the Banach Fixed Point Theorem for Mappings in (R,
More informationGeneralized Derivation on TM Algebras
International Journal of Algebra, Vol. 7, 2013, no. 6, 251-258 HIKARI Ltd, www.m-hikari.com Generalized Derivation on TM Algebras T. Ganeshkumar Department of Mathematics M.S.S. Wakf Board College Madurai-625020,
More informationA Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials
Applied Mathematical Sciences, Vol. 8, 2014, no. 35, 1723-1730 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4127 A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating
More informationResearch Article On Prime Near-Rings with Generalized Derivation
International Mathematics and Mathematical Sciences Volume 2008, Article ID 490316, 5 pages doi:10.1155/2008/490316 Research Article On Prime Near-Rings with Generalized Derivation Howard E. Bell Department
More informationOn Uniform Limit Theorem and Completion of Probabilistic Metric Space
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 455-461 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4120 On Uniform Limit Theorem and Completion of Probabilistic Metric Space
More informationMorera s Theorem for Functions of a Hyperbolic Variable
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 32, 1595-1600 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.212354 Morera s Theorem for Functions of a Hyperbolic Variable Kristin
More informationImprovements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear
More informationAn Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh
International Mathematical Forum, Vol. 8, 2013, no. 9, 427-432 HIKARI Ltd, www.m-hikari.com An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh Richard F. Ryan
More informationOn a Principal Ideal Domain that is not a Euclidean Domain
International Mathematical Forum, Vol. 8, 013, no. 9, 1405-141 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.013.37144 On a Principal Ideal Domain that is not a Euclidean Domain Conan Wong
More informationAn Envelope for Left Alternative Algebras
International Journal of Algebra, Vol. 7, 2013, no. 10, 455-462 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.3546 An Envelope for Left Alternative Algebras Josef Rukavicka Department
More informationON HOCHSCHILD EXTENSIONS OF REDUCED AND CLEAN RINGS
Communications in Algebra, 36: 388 394, 2008 Copyright Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870701715712 ON HOCHSCHILD EXTENSIONS OF REDUCED AND CLEAN RINGS
More informationTHE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter
THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,
More informationOrthogonal Derivations on Semirings
International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 13, 645-651 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.49100 Orthogonal Derivations on Semirings N.
More informationOn Symmetric Bi-Multipliers of Lattice Implication Algebras
International Mathematical Forum, Vol. 13, 2018, no. 7, 343-350 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2018.8423 On Symmetric Bi-Multipliers of Lattice Implication Algebras Kyung Ho
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationNIL DERIVATIONS AND CHAIN CONDITIONS IN PRIME RINGS
proceedings of the american mathematical society Volume 94, Number 2, June 1985 NIL DERIVATIONS AND CHAIN CONDITIONS IN PRIME RINGS L. O. CHUNG AND Y. OBAYASHI Abstract. It is known that in a prime ring,
More informationFinite Groups with ss-embedded Subgroups
International Journal of Algebra, Vol. 11, 2017, no. 2, 93-101 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7311 Finite Groups with ss-embedded Subgroups Xinjian Zhang School of Mathematical
More informationOre extensions of Baer and p.p.-rings
Journal of Pure and Applied Algebra 151 (2000) 215 226 www.elsevier.com/locate/jpaa Ore extensions of Baer and p.p.-rings Chan Yong Hong a;, Nam Kyun Kim b; 1, Tai Keun Kwak c a Department of Mathematics,
More informationSurjective Maps Preserving Local Spectral Radius
International Mathematical Forum, Vol. 9, 2014, no. 11, 515-522 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.414 Surjective Maps Preserving Local Spectral Radius Mustapha Ech-Cherif
More informationRegular Weakly Star Closed Sets in Generalized Topological Spaces 1
Applied Mathematical Sciences, Vol. 9, 2015, no. 79, 3917-3929 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.53237 Regular Weakly Star Closed Sets in Generalized Topological Spaces 1
More informationON WEAK ARMENDARIZ RINGS
Bull. Korean Math. Soc. 46 (2009), No. 1, pp. 135 146 ON WEAK ARMENDARIZ RINGS Young Cheol Jeon, Hong Kee Kim, Yang Lee, and Jung Sook Yoon Abstract. In the present note we study the properties of weak
More informationOn Strongly Clean Rings
International Journal of Algebra, Vol. 5, 2011, no. 1, 31-36 On Strongly Clean Rings V. A. Hiremath Visiting professor, Department of Mathematics Manglore University, Mangalagangotri-574199, India va hiremath@rediffmail.com
More informationLinearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 9, 2015, no. 58, 2889-2900 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121002 Linearization of Two Dimensional Complex-Linearizable Systems of
More informationA GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis
International Electronic Journal of Algebra Volume 20 (2016) 111-135 A GENERAL HEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUAIVE RING David F. Anderson and Elizabeth F. Lewis Received: 28 April 2016 Communicated
More informationNon Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical Systems
Applied Mathematical Sciences, Vol. 12, 2018, no. 22, 1053-1058 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.87100 Non Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical
More informationOn Regular Prime Graphs of Solvable Groups
International Journal of Algebra, Vol. 10, 2016, no. 10, 491-495 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.6858 On Regular Prime Graphs of Solvable Groups Donnie Munyao Kasyoki Department
More information