Abstract

Let R be a ring, and let N and C denote the set of nilpotents and the center of R, respectively. R is called generalized periodic if for every x ∈ R\(N ⋃ C), there exist distinct positive integers m, n of opposite parity such that xn − xm ∈ N ⋂ C. We prove that a generalized periodic ring always has the set N of nilpotents forming an ideal in R. We also consider some conditions which imply the commutativity of a generalized periodic ring.

Highlights

  • Throughout the paper, R will denote a ring, N the set of nilpotents, C the center, J the Jacobson radical, and C(R) the commutator ideal of R

  • We formally state the definition of a generalized periodic ring

  • We prove that the set of nilpotents in a generalized periodic ring R is always an ideal in R

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Summary

Introduction

Throughout the paper, R will denote a ring, N the set of nilpotents, C the center, J the Jacobson radical, and C(R) the commutator ideal of R. A ring R is called generalized periodic if for every x in R, x N u C, we have x* x" N n C, for some positive integers m, n of opposite parity. We prove that the set of nilpotents in a generalized periodic ring R is always an ideal in R. We consider conditions which imply the commutativity of a generalized periodic ring.

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