Periodic rings with commuting nilpotents

Abstract

Let R be a ring (not necessarily with identity) and let N denote the set of nilpotent elements of R. Suppose that (i) N is commutative, (ii) for every x in R, there exists a positive integer k=k(x) and a polynomial f(λ)=fx(λ) with integer coefficients such that xk=xk+1f(x), (iii) the set In={x|xn=x} where n is a fixed integer, n>1, is an ideal in R. Then R is a subdirect sum of finite fields of at most n elements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (taking n=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume that In is a subring of R.

Keywords

• Boolean ring
• subdirect sum
• subdirectly irreducible.