On R-Strong Jordan Ideals

Transcription

1 International Journal of Algebra, Vol. 3, 2009, no. 18, On R-Strong Jordan Ideals Anita Verma Department of Mathematics University of Delhi, Delhi 1107, India Abstract. R-strong Jordan Ideals have been defined. Examples have been given to show their existence. It has been proved that the sum of two R-strong Jordan Ideals is an R-strong Jordan Ideal. Also, it has been proved that the intersection of an arbitrary number of R-strong Jordan ideals is an R-strong Jordan ideal. Further, we prove that the product of two R-strong Jordan ideals is also an R-strong Jordan ideal. Finally, a set A V associated with an R-strong Jordan ideal V has been defined and sufficient condition, under which A V V is an R-strong Jordan ideal, has been given. Mathematics Subject Classification: 16A66, 16A72 Keywords: Ideals, Jordan Ideal, R-strong Jordan Ideal 1. Introduction Throughout the paper, we assume that R is a non-commutative ring, the symbol J denotes the Jordan ideal of R. A ring R is said to be prime if for a, b R, arb = (0) implies a =0orb = 0. An additive subgroup J of R is said to be a Jordan ideal of R if ur + ru J for all u J, r R. Forx, y R, by [x, y], we mean xy yx. An additive subgroup U of R is said to be a Lie ideal of R if [a, r] U, for all a A, r R. One may observe that if char R = 2, then Jordan ideal and Lie ideal of R are same. Also every ideal of R is Jordan ideal of R but converse need not be true. Further, one may verify that the intersection of an arbitrary number of R-strong Jordan ideals of R is also a Jordan ideal of R.

2 898 A. Verma 2. Jordan ideals It may be observed that if I 1 and I 2 are two ideals of R, then I 1 + I 2 is an ideal of R. However, it is not true in case of Jordan ( ideals. ) Indeed, let R be a 1 1 ring of 2 2 matrices over integers and let a = R. Then {( ) x y ar = x, y Z} Since ( )( ) ( )( ) x y t s t s x y + u v u v ( ) ( ) xt + yu xs + yv tx ty = + ar, ux uv ar is not a Jordan ideal of R. Similarly, Ra is not a Jordan ideal of R. Also, one may verify that ar + Ra is a Jordan ideal of R. Lemma 2.1. Let R be a ring with unity and 2R = R. IfJ is the Jordan ideal of R and 1 J. Then J = R. Proof. Obviously J R. Let r R be any element. Since 1 J and J is Jordan ideal of R, 1 r + r 1 J. This gives 2r J. Hence r J. Lemma 2.2. If J is a Jordan ideal of R and b J, then [[x, y],b] J, for all x, y R. Proof. Let b J. Since J is Jordan, xb + bx and yb + by J. Also, since J is an additive subgroup of R, (bx + xb)y + y(bx + xb) J and (by + yb)x + x(by + yb) J. This give b[x, y] [x, y]b J. Hence [[x, y],b] J. 3. R-Strong Jordan Ideals Throughout this section by ring R, we mean a prime ring. Definition 3.1 ([1]). Let R be a prime ring. A Jordan ideal V of R, is said to be R-strong Jordan ideal of R, ifavb V, for all v V and for all a, b R. Towards the existence of R-strong Jordan ideals, we give the following example.

3 On R-strong Jordan ideals 899 {( ) {( ) x 0 a 0 Example 3.2. (i) Let R = x, y Z} and let V = a Z,a 0}. 0 y ( ) x 0 Then V is R-strong Jordan ideal of R. Indeed, let if X = R and 0 y ( ) a 0 Y = V, then ( ) xa + ax 0 XY + YX = V ( ) xax 0 Also, XY X = V. Hence V is an R-strong Jordan ideal of R. a (ii) Let R = 0 d e a, d, e, f Z and f a V = 0 a Z,a 0. 0 Then V is a R-strong Jordan ideal of R. Note. R-strong Jordan ideal of R is a Jordan ideal of R. Theorem 3.3. If V 1 and V 2 are two R-strong Jordan ideals of R, then V 1 +V 2 is also R-strong Jordan ideal of R. Proof. Clearly V 1 +V 2 is a Jordan ideal of R. Let x V 1 +V 2 and a, b R. Then x = y + z, y V 1,z V 2. Since V 1 is R-strong Jordan ideal of R, for y V 1 and a, b R, ayb V 1. Similarly, azb V 2. Also, axb = ayb + azb V 1 + V 2. Hence V 1 + V 2 is an R-strong Jordan ideal of R. Theorem 3.4. Let {V t : t T, where T is an indexed set } be a family of R-strong Jordan ideals of R. Then V t is an R-strong Jordan ideal of R. t T Proof. Let V = V t. Let x V and a, b R. Since x V, x V t, for all t T t T.Nowx V t and V t is R-strong Jordan ideal, therefore axb V t, for all t T. Hence axb V t = V. t T Remark 3.5. Union of two R-strong Jordan ideals need not be an R-strong Jordan ideal. Indeed, if {( ) {( ) x 0 R = x, y Z}, V 0 y 1 = a, b Z} a b

4 900 A. Verma {( ) a b and V 2 = a, b Z}, then both V 1 and V 2 are R-strong Jordan ideals of R. But V 1 V 2 is not even a Jordan ideal of R. Indeed, ( )( ) ( )( ) ( ) x 0 a b x 0 xa bx + = V 0 y a b 0 y ax by 1 V 2. Regarding product of two R-strong Jordan ideals, we give the following result. Theorem 3.6. Let R be a ring with unity. If V 1 and V 2 are R-strong Jordan ideals of R, then V 1 V 2 is also an R-strong Jordan ideal of R. Proof. Note that { n } V 1 V 2 = a i b i a i V 1,b i V 2,n Z clearly V 1 V 2 is a Jordan ideal of R. Let x V 1 V 2 and r R. Then x = n a ib i, a i V 1, b i V 2. Now a i V 1, r R and V 1 is R-strong Jordan ideal, therefore, ra i r V 1. Similarly, rb i r V 2,, 2,...,n. Now ( n n n (3.3.1) r a i b i )r = (ra i r rb i r)+ (ra i ra i r)(b i r + rb i r) Since r 1 a i r 2 V 1, for all r 1,r 2 R and i =1, 2,...,n. Taking r 1 = r 1,r 2 = r, we get (ra i r a i r) V 1,, 2,...,n. Similarly b i r + rb i r V 2.So n (ra i ra i r)(b i r + rb i r) V 1 V 2. Thus (3.1) gives, r( n a ib i )r V 1 V 2. Lemma 3.7. Let V be an R-strong Jordan ideal of R, where R is a ring with unity. If v V and a, b R, then abv + vba V. Proof. Let avb and bva V. Then avb (3.3.2) + bva V Now since V is a Jordan ideal, a(avb + bva)+(avb + bva)a V. Therefore, by (3.2), avba + abva V. Replacing a by (a 1), we get (avb vb)(a 1) + (abv bv)(a 1) V This gives vba abv V. Hence vba + abv V. Let V be a R-strong Jordan ideal of R. Ifa, b R, we associate V with the set A V = {b R : ab + ba V, for all a R}

5 On R-strong Jordan ideals 901 Theorem 3.8. If V is an R-strong Jordan ideal of R, then A V is an R-strong Jordan ideal of R. Proof. Let x A V and r R. Since x A V,xr+ rx V. Also, since V is a Jordan ideal of R, (xr + rx)y + y(xr + rx) V. This gives xr + rx A V. Hence A V is a Jordan ideal of R. Let b B J, x, y R. Since b B J, x, y R, xb + bx, yb + by V. Since V is an R-strong Jordan ideal, y(by + yb)y V. This implies that yby (3.3.3) 2 + y 2 by V. Similarly, x(by + yb)y V and y(by + yb)x V. This gives xby 2 + xyby V and ybyx + y 2 bx V. Hence, by (3.3) x(yby)+(yby)x V. Hence A V is R-strong. Theorem 3.9. If R is a ring with 2R = R and V is an R-strong Jordan ideal of R, then A V V is a non-zero right ideal of R. Proof. Note that A V V (0). Let b A V V, x, y R. Then bx + xb V. So, bx + xb A V. Hence bx + xb A V V.Now xb + bx = xb + bx + xb xb = bx xb +2xb = bx xb + xb ( 2R = R) = bx A V V Since x R is arbitrary, bx A V V, for all x R. Hence A V V is a non-zero right ideal of R. Theorem If e is an idempotent and V is a Jordan ideal of R, then ev e is an ere-strong Jordan ideal of R. Proof. Let x ev e and r ere. Then xr + rx = e(vr 1 + r 1 v), v V,r 1 R ev e ( r 1 e = r 1 = er 1 ) Again, let u ev e and x, y ere. Then xuy = e(xvy)e ev e. References [1] A. J. Karam, Strong Lie ideals, Pacifie Journal of Mathematics, 43 (1) (1972), [2], Concerning strong Lie ideal, Proc. Amer. Math. Soc., 11 (1960), [3] I. N. Herstein, Topics in Ring Theory, Univ. of Chicago, Chicago III, 1965, revised edition, 1969.

6 902 A. Verma [4], Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), [5], Lie and Jordan systems in simple rings with involution, Amer. J. Math., 78 (1956), [6] R. Awtar, Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc., 90 (1) (1982), [7] W. E. Baxter, On rings with proper involution, Pacifie Journal of Mathematics, 27 (1), Received: March, 2009