Abstract

COMMUTATIVITY OF SOME SPECIAL CLASSES OF RING

Moharram A. Khan & Abdu Madugu

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Abstract In this paper, we prove the commutativity of certain rings such as rings with unity 1 and one-sided s-unital rings satisfying one of the following ring properties as defined below: (P1) For each x? R, there exist polynomials f(x),g(x),h(x)?Z[x] such that x^p [x^m,y] y^q=?g(y) [x,f(y)]^k h(y)?y?R, fixed integers p?0,q?0, k?1,t?2 and m>1. (P2) For each x,y? R, there exist polynomials f(x),g(x),h(x),f ?(x),g ?(x),h ?(x)?Z[x] , p=p(x,y)?0,q=q(x,y)?0,m=r(x,y)>1,n=n(x,y)>1,k?1,t?2 are integers with (m,n)=1 and R satisfies x^p [x^m,y] y^q=? g(y)?[ ?x,y?^t f(y)]?^k h(y) and? x?^q [x^n,y] y^p=?g ?(x)?[?x,y?^t f ?(y)]?^k h ?(y). In addition, commutativity of rings is established under different set of constraints on integral exponents. Finally, we conclude with some problems in rings. Keywords and phrases: Commutativity, commutators, rings with unity 1, one-sided s-unital rings.

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