Gamma Rings of Gamma Endomorphisms

Transcription

1 Annals of Pure and Applied Matheatics Vol. 3, No.1, 2013, ISSN: X (P), (online) Published on 18 July Annals of Md. Sabur Uddin 1 and Md. Sirajul Isla 2 1 Departent of Matheatics Carichael College, Rangpur-5400 Bangladesh 2 Departent of Matheatics Bangabandhu Sheikh Mujibur Rahan Science & Technology University Gopalganj, Bangladesh Received 10 July 2013; accepted 16 July 2013 Abstract. In this paper we study Γ-rings of Γ-endoorphiss. Soe properties of Irreducible Γ-rings are developed by the help of Γ-endoorphiss. Keywords: Gaa Endoorphiss, Irreducible Gaa rings, Gaa hoorphiss, Gaa Isoorphis. AMS Matheatics Subject Classification (2010): 16N60, 16W25, 16U80 1. Introduction As a generalization of ring theories, the concept of Γ-rings was first introduced by N. Nobusawa [6]. After wards Barnes [1] generalized the notion of Nobusawa s Γ-rings and gave a new definition of a Γ-ring. Now a days, Γ-rings eans the Γ-rings due to Barnes and other Γ-rings are known as ΓN -rings i.e., gaa rings in the sense of Nobusawa. Many Matheaticians worked on Γ-rings and obtained soe fruitful results which are the generalizations of that of the classical ring theories. W.E. Barnes [1] introduced the notation of Γ-hooorphiss, Prie and Priary ideals, -systes and the radical of an ideal for Γ-rings. In classical ring theories N.H. McCoy [5] studied rings of endoorphiss. In this paper we generalized the results of N.H. McCoy [5] into Γ-rings of Γ- endoorphiss. We also developed soe characterizations of Γ -rings by the help of Γ-endoorphiss. 2. Preliinaries Gaa Ring. Let M and Γ be two additive abelian groups. Suppose that there is a apping fro M Γ M M (sending (x, α, y) into xαy) such that (i) (x + y)αz = xαz + yαz x(α + β)z = xαz + xβz xα(y + z) = xαy + xαz (ii) (xαy)βz = xα(yβz), 94

2 where x, y, z M and α, β Γ. Then M is called a Γ-ring in the sense of Barnes [1]. Sub-Γ-ring. Let M be a Γ-ring. A non-epty subset S of a Γ-ring M is a sub-γ-ring of M if a, b S, then a b S and aγb S for all γ Γ. Ideal of Γ-rings. A subset A of the Γ-ring M is a left (right) ideal of M if A is an additive subgroup of M and MΓA = {cαa c M, α Γ, a A}(AΓM) is contained in A. If A is both a left and a right ideal of M, then we say that A is an ideal or two-sided ideal of M. If A and B are both left (respectively right or two-sided) ideals of M, then A + B = {a + b a A, b B} is clearly a left (respectively right or two-sided) ideal, called the su of A and B. We can say every finite su of left (respectively right or two-sided) ideal of a Γ-ring is also a left (respectively right or two-sided) ideal. It is clear that the intersection of any nuber of left (respectively right or two-sided) ideal of M is also a left (respectively right or two-sided) ideal of M. If A is a left ideal of M, B is a right ideal of M and S is any non-epty subset of M, then the set, AΓS = { n i= 1 a i γs i a i A, γ Γ, s i S, n is a positive integer} is a left ideal of M and SΓB is a right ideal of M. AΓB is a two-sided ideal of M. If a M, then the principal ideal generated by a denoted by a is the intersection of all ideals containing a and is the set of all finite su of eleents of the for na + xαa + aβy + uγaµv, where n is an integer, x, y, u, v are eleents of M and α, β, γ, µ are eleents of Γ. This is the sallest ideal generated by a. Let a M. The sallest left (right) ideal generated by a is called the principal left (right) ideal a ( a ). Unity eleent of a Γ-ring. Let M be a Γ-ring. M is called a Γ-ring with unity if there exists an eleent e M such that aγe = eγa = a for all a M and soe γ Γ. We shall frequently denote e by 1 and when M is a Γ-ring with unity, we shall often write 1 M. Note that not all Γ-rings have an unity. When a Γ-ring has an unity, then the unity is unique. Γ -hooorphis. Let M and S be two Γ-rings. A apping θ of a Γ-ring M into a Γ- ring S is said to be a Γ-hooorphis of M into S if addition and ultiplication are preserved under this apping, that is, if ab Μ,, α Γ (i) ( a+ b) θ = aθ + bθ (ii) ( aα b) θ = ( aθ) α( bθ). If θ is one-one and onto then θ is called a Γ-isoorphis fro M into S. Kernel of Γ-hooorphis. If θ is a hooorphis of a Γ-ring M into a Γ-ring S, 1 then (0) θ, that is, the set of all eleents a of M such that aθ = 0 (the zero of S), is called the kernel of the Γ-hooorphis θ. 95

3 Md. Sabur Uddin and Md.Sirajul Isla 3. Definition 3.1. Mapping a : Μ M of the Γ -ring M into itself is called a Γ - endoorphis of M if for x, y Μ, α Γ, then ( x + ya ) = xa+ ya (1) ( xα ya ) = xaα ya. (2) If a is a Γ -endoorphis of M, then 0a= 0,0 Μ,( x) a= ( xa) and ( xα y)0= 0, x, y Μ, α Γ. It is to be understood that equality of appings is the usual equality of appings, in other words a = b for Γ -endoorphiss a and b of M eans that xa = xb for every x Μ. Let us denote by, the set of all Γ -endoorphis of the Γ -ring M. We now define ultiplication and addition on the set as follows: where it is understood that a and b are eleents of : xa ( αb) = ( xa) αb, x Μ, α Γ (3) xa ( + b) = xa+ xb, x Μ. (4) Ofcourse (3) is just the usual definition of ultiplication of appings of any set into itself, whereas (4) has eaning only because we already have an operation of addition defined on M. The fact that aα band a+b are indeed Γ -endoorphis of M and therefore eleents of follows fro the following siple calculations in which x and y are arbitrary eleents in M: ( x + y)( aα b) = (( x+ y) a) αb by(3) = ( xa+ ya) αb by (1) = ( xa) αb+ ( ya) αb by(1) = xa ( αb) + ya ( αb) by(3); and ( x + y)( a+ b) = ( x+ y) a+ ( x+ y) b by(1) = xa+ ya+ xb+ yb by(1) = xa+ xb+ ya+ yb = x( a+ b) + y( a+ b) by(4). Now that we have addition and ultiplication defined on the set, we ay state the following theore. Theore 3.2. With respect to the operations (4) and (3) of addition and ultiplications the set of all Γ -endoorphiss of the Γ -ring M is a Γ -ring with unity. Proof. (i) Let abc,,, α Γ and x Μ, then x(( a+ b) ) = x( a+ b) = ( xa+ xb) = ( xa) + ( xb) c) + x( bα c) c+ bα c). Hence ( a+ b) = a+ b. Now xa ( ( α + β) c) = ( xa)( α + β) c, ac,, α, β Γ, x Μ 96

4 = ( xa) + ( xa) βc c) + x( aβc) c) + aβc). Thus a( α + β) c= a+ aβc. Again x( aα ( b+ c)) = ( xa) α( b+ c), a, b, c, α Γ, x Μ = ( xa) αb+ ( xa) b) + x( aα c) b+ aα c). Hence aα ( b+ c) = aαb+ a. (ii) x(( aα b) βc) = ( xa ( αb)) βc, abc,, α, β Γ = (( xa) αb) βc = ( xa) α( bβc) ( bβc)). Hence ( aα b) βc= aα( bβc). (iii) For all a, then exists unity eleent 1 such that x(1 αa) = (( x1) α) a= xa, α Γ, x Μand x( aα1) = (( xa) α)1 = xa. Thus x( aα1) = x(1 αa) = xa. Hence aα1= 1α a = a. Thus satisfies all the conditions of a Γ -ring. Hence is a Γ - ring with unity. Theore 3.3. Let be the Γ -ring of all Γ -endoorphiss of the Γ -ring M. If a, then a has an inverse in if and only if a is a one-one apping of M onto M. Proof. Suppose first that a has an inverse b in, so that aα b= bαa= 1, α Γ. Then for each x Μ, we have ( xb) αa= x( bα a) = x1= x and a is clearly a apping onto M. Moreover, if x1, x2 Μ such that xa 1 = xa 2, then x1= x11 = x1( aα b) = ( xa 1 ) αb= ( x2a) αb= x2( aα b) = x21 = x2. This shows that the apping is a one-one apping. Conversely, let us assue that the Γ -endoorphis a is a one-one apping of M onto M, so that every eleent of M is uniquely expressible in the for xa, x Μ. We ay therefore define a apping b of M into M as follows: (( xa) α) b = x, x Μ, α Γ. Since, if x, y Μ, then (( xa+ ya) α) b= ((( x+ y) a) α) b= x+ y= (( xa) α) b+ (( ya) α) b and (( xaα ya) α) b= (( xαy) aα) b= ( xαy) = (( xa) α) bα(( ya) α) b. We see that b is a Γ - endoorphis of M. Moreover ( xa) αb b) = x1= x for every x in M and hence aα b= 1. Finally if x Μ, then (( xa) α)( bαa) = ( x( aαb)) αa= ( x1) αa= x(1 αa) = xa. Thus is equivalent to the stateent that yb ( α a) = yfor every y Μ. Hence bα a = 1 and b is the inverse of a in. 97

5 Md. Sabur Uddin and Md.Sirajul Isla Suppose now that we start with a given Γ -ring M and let be the Γ -ring of all Γ - endoorphiss of M. If a is a fixed eleent of M, then the apping x xα a, α Γ, x Μ of M into M is called the right ultiplication by a. It will be convenient to denote this right ultiplication by a, that is, the apping a is defined by xα a = xαa, x Μ, α Γ. It is clear that a is a Γ -endoorphis of M and therefore a for each a Μ. If ab Μ,, then a + b and aαb, α Γ are defined in. Moreover we observe that for each x in M, xα ( a+ b) = xα( a+ b) = xαa+ xαb = xα a + xα b = xα ( a + b) and xα ( aαb) = xα( aαb) = ( xαa) αb = ( xα a) αb = ( xα a) αb = xα ( aαb). Thus we ( a+ b) = a + b (5) ( aαb) = aαb Hence a + b and aαb, α Γ we theselves right ultiplications and the set S of all right ultiplications is a sub- Γ -ring of. Moreover, the relations (5) shows that the apping a a, a Μ, is a Γ -hooorphis of M onto S. If a is in the kernel of this Γ -hooorphis, then xα a = 0 for every eleent x Μ and α Γ. If M happens to have a unity, this iplies that a = 0 and in this case the kernel is certainly zero and the Γ -hooorphis is a Γ -isoorphis. We have therefore established the following result. Lea 3.4. If the Γ -ring has a unity, then M is Γ -isoorphic to the Γ -ring of all its right utiplications. For our purposes, the iportance of this Lea is that it leads atost iediately to the following result. Theore 3.5. Every Γ -ring M is Γ -isoorphic to a Γ -ring of Γ -endoorphiss of M. If is a Γ -ring of Γ -endoorphiss of a Γ -ring M and U is a sub- Γ -ring of M. U = xa x U, a. If U consists of a single We naturally define U as follows: { } y, that is y = { yaa }. Definition 3.6. If is a Γ -ring of Γ -endoorphiss of a Γ -ring M and W is a sub- Γ - ring of M such that W W, then W ay be called a -sub- Γ -ring of M. Clearly, the zero sub- Γ ring denote siply by 0 and the entire Γ -ring M are always -sub- Γ -rings. We shall be particularly interested in the case in which there are no -sub- Γ -rings except these two trivial ones. eleent y of M, we write y instead of { } Definition 3.7. Let be a nonzero Γ -ring of Γ -endoorphiss of the Γ -ring M. If the only -sub- Γ -rings W of M are W = 0 and W = M. We say that is an irreducible Γ -ring of Γ -endoorphiss of M. 98

6 We ay reark that if M is the zero Γ -ring (having only the zero eleent) the only Γ -endoorphis of M is the zero Γ -endoorphis. Since in the proceeding definition is required to be a non-zero Γ -ring of Γ -endoorphiss of M, when we speak of an irreducible Γ -ring of Γ -endoorphiss of M it is iplicit that M ust have nonzero eleents. We shall next prove the following useful result. Lea 3.8. If is a nonzero Γ -ring of Γ -endoorphiss of the Γ -ring M, then is an irreducible Γ -ring of Γ -endoorphiss of M if and only if x =Μ for every nonzero eleent x of M. Proof. One part is essentially trivial. For if x =Μ for every nonzero eleent x of M, it is clear that M is the only nonzero -sub- Γ -ring of M and is therefore irreducible. Conversely, let us assue that is an irreducible Γ -ring of Γ -endoorphiss of M and that x in an arbitrary nonzero eleent of M. It is easily verified that x is a sub- Γ -ring of M and since ( x Γ ) x, it is a -sub- Γ -ring of M. It follows that x = 0 or x =Μ. Suppose that x = 0 and let < x > be the sub- Γ -ring of M generated by x. Then < x > = 0 and therefore < x > is an -sub- Γ -ring of M. Since x < x > and x 0, we ust have < x >=Μ and therefore Μ = 0. However, this is ipossible since M has nonzero eleents and the assuption that x = 0 has led to a contradiction. Hence x =Μ and the proof is copleted. Theore 3.9. Let be an irreducible Γ -ring of Γ -endoorphiss of the Γ -ring M. If A is a nonzero ideal in, then A also is an irreducible Γ -ring of Γ -endoorphiss of M. Proof. Let x be an arbitrary nonzero eleent of M. Since ΑΓ Α. We have that ( xαγ = ) x( ΑΓ ) xα, so that xα is a -sub- Γ -ring of M. Since is irreducible, it follows that xα= 0 or xα =Μ. Suppose that xα = 0. Then using Lea 3.8, we have ΜΑ = ( x ) Γ xα = 0, which is ipossible since A is assued to have nonzero eleents. Accordingly we ust have xα =Μ. The sae lea, now applied to the Γ -ring A, shows that A is indeed an irreducible Γ -ring of Γ - endoorphis of M and the proof is copleted. REFERENCES 1. W.E. Barnes, On the gaa rings of Nobusawa, Pacific J. Math., 18 (1966) C.W. Bitzer, Inverse in rings with unity, Aer. Math. Monthly, 70 (1963), R.E Johnson and F.Kiokeeister, The endoorphiss of the total operator doain of an infinite odule, Trans. Aer. Math. Soc., 62 (1947) N.H.Mc Coy, Subdirectly irreducible coutative rings, Duke, Math. J., 12 (1945) N.H.Mc Coy, The Theory of Rings, The Macillan Copany, New York (1967). 6. N.Nobusawa, On a generalization of the ring theory, Osaka J. Math., 1(1964)