Abstract

Herstein proved that a ring R which satisfies the identity , where n is a fixed positive integer greater than 1, must have nil commutator ideal. Abu-Khuzam provethat if n is a fixed positive integer greater than 1 and R is an n (n-1) – torsion free ring with unity such that for all x,y in R, then R is commutative. In Gupta proved that if R is a semi prime ring satisfying for all x,y in R, where Z(R) in the center of R, then R is commutative. In it is proved that a semi prime ring R such that for each x in R there exists a positive integer such that and for all y in r then R is commutative. In this direction we prove that if R is an n (n+1)-torsion free periodic ring such that and or and if the set of nilpotent elements of R is commutative. Then R is commutative. We know that R is periodic if for every x in R, there exist distinct positive integers m and n such that . By a theorem of Chancron , R is periodic if and only if for each x in R, there exists a positive integer k = k(x) and a polynomial with integer coefficients such that . Throughout this section R is an associative ring, Z(R) denotes the center of R and N denotes the set of nilpotent elements of R.

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References

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