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A Result for Semi Simple Ring International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019 A Result for Semi Simple Ring 1 2 Jameel Ahmad Ansari, Dr. Haresh G Chaudhari 1Assistant Professor, 2Assistant Professor, 1Department of Engineering Sciences, 1 Vishwakarma University, Pune, Maharashtra, India 2Department of Mathematics, 2MGSM’S Arts, Science and Commerce College, Chopda, Jalgaon, Maharashtra, India Abstract If for a semi simple ring 푅, with any elements 푎, 푏 in 푅 there exist positive integers 푝 = 푝(푎, 푏) and 푞 = 푞(푎, 푏) such that [ 푏푎푏 푝 , (푎푏)푞 + (푏푎)푞 ] = 0. then 푅 is commutative. Key words - Skew field, Simple ring, Semi simple ring & Primitive ring. I. INTRODUCTION M. Ashraf [1] proved: Let 푅be a semi simple ring. Suppose that given 푥, 푦in 푅 there exist positive integers 푚 = 푚(푥, 푦)and 푛 = 푛(푥, 푦)such that [xm , 푥푦 푛 ] = [(yx)n , 푥푚 ] then 푅is commutative. We have weakened the M. Ashraf identity to make a stronger generalization of M. Ashraf [1]. This reads as follows : A. THEOREM: Let 푅be a semi simple ring. Suppose that given 푥, 푦in 푅, there exist positive integers 푝 = 푝(푎, 푏)and 푞 = 푞(푎, 푏)such that [ 풃풂풃 풑, ab q + 푏푎 푞 ] = 0. Then 푅is commutative. Throughout this paper 푅is taken as an associative ring. 푥, 푦 = 푥푦 − 푦푥 and 푥표푦 = 푥푦 + 푦푥for every pair 푥, 푦in 푅. B. Division Ring (Skew Fields) A division ring, also called a skew field, is a non commutative ring with unity in which every non zero element has inverse. C. Simple Ring A simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. e.g: Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. In fact a division ring is also a simple ring. D. Semi Simple Rings A ring in which unit element does not equal to zero element and which is semi simple as (left) module over itself is semisimple ring. E. Primitive Rings A ring R is said to be a left primitive ring if and only if it has a simple left R-module. A right primitive ring is defined similarly with right R-modules. II. PREPARATORY RESULTS This section contains those lemmas which are required to prove the theorem. LEMMA 2.1: ([3], Theorem 1) : Let 푅be a ring without nonzero nil right ideals. Suppose that given 푎, 푏in 푅. there exist positive integers 푝 = 푝 푥, 푦 , 푞 = 푞(푥, 푦) > 1 and 푡 = 푡(푥, 푦) > 1 such that 푥푝 , 푥푞 , 푦푡 = 0. Then 푅 is commutative. LEMMA 2.2: Let 푥, 푦, 푧be the elements of a ring 푅. (i) If 푥, 푦 = 0then [푥, 푦표푧] = 푦표[푥, 푧]. (ii) 푥표[푥, 푦] = [푥2, 푦]. PROOF: Proof of this lemma is straight forward. LEMMA 23: Let 푹 be a Skew field. Suppose that given 풂, 풃in 푹,there exist positive integers 풑 = 풑(풂, 풃) and 풒 = 풒(풂, 풃)such that [(풃풂풃)풑, (풂풃)풒 + (풃풂)풒] = ퟎ. Then 푹 is commutative. ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 214 International Journal of Mathematics Trends and Technology (IJMTT) - Volume 65 Issue 3 - March 2019 PROOF: Let 풃 ≠ 0, then by hypothesis there exist positive integers풑 = 풑(풃−ퟏ풂풃−ퟏ, 풃)and 풒 = 풒(풃−ퟏ풂풃−ퟏ, 풃) such that the identity given in the hypothesis reduces to [ b풃−ퟏ풂풃−ퟏ풃 p, 풃−ퟏ풂풃−ퟏ풃 풒 + (풃풃−ퟏ풂풃−ퟏ)풒] = ퟎ i.e. [풂풑, (풃−ퟏ 풂)풒 + (풂풃−ퟏ)풒] = 0 (1) Replacing풃by 풂풃−ퟏin (1).we get[풂풑, 풃풒 + 풂풃풒풂−ퟏ ] = ퟎ. This on simplification and consequently multiplying by 풂 on the right hand side yields, 풂풑풃풒 + 풂풑+ퟏ풃풒 − 풃풒풂풑+ퟏ − 풂풃풒풂풑 = 0 (2) Therefore 풂풑, 풂풐풃풒 = 0Now by lemma 2.2(i), we obtain 풂풐[풂풑, 풃풒]when 풂, 풂풐 풂풑, 풃풒 = ퟎNow using parts (i) and (ii) consecutively of lemma 2.2, we have 풂ퟐ, 풂풑, 풃풒 = ퟎ.Hence by lemma 2.1, 푹is commutative. III. PROOF OF THE THEOREM Suppose that 푹is a semi simple ring such that for all 풙, 풚in 푹there exist positive integers 풑 = 풑(풂, 풃) and 풒 = 풒(풂, 풃) fo which [ 풃풂풃 풑, 풂풃 풒 + (풃풂)풒] = 0 (A) A semi simple ring is isomorphic to a sub direct sum of primitive rings. Further the identity (A) satisfied by a ring is also satisfied by all its sub rings and homomorphic images. So to prove the theorem for semi simple rings it suffices to prove it for primitive rings. Now every Skew field is primitive (5.20 Ex. 5, [4]), but a primitive ring need not be a Skew field, in general. Here in the present case, the primitive ring satisfying the identity (A) is necessarily a Skew field. Because a primitive ring by Jacobson Density Theorem is isomorphic to a complete matrix ring 푫풕where D is a Skew field and 풕 > 1. But we observe that identity (A) is not satisfied by any complete matrix ring 푫풕e.g. a consideration of 풂 = 풆ퟏퟏ + 풆ퟏퟐand 풃 = 풆ퟏퟏfor 풕 = 2 leads that (A) is notsatisfied. Thus our primitive ring 푹 is necessarily a Skew field. Hence theorem follows from lemma 2.3. REFERENCES [1] QUADRI. M.A and KHAN. M.A; A theorem on commutativity of semisimple rings, soochow Journal of Math. 11 (1985), 97-99. [2] JACOBSON. N; Structure of rings, Amer. Math. Society, Colloq. Publ. 37, Providence 1964. [3] HONGAN, M. and TOMINAGA H; Somecommutativity theorem for semiprime rings, Hokkaido Math. J. 10 (1981), 271-277. [4] McCOY, N. H; Theory of rings, MacMillan Press, New York. ISSN: 2231 – 5373 http://www.ijmttjournal.org Page 215 .
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