Nil Subrings of Goldie Rings are Nilpotent

Abstract

Herstein and Small have shown ( 1 ) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring. Definition. L is a left annihilator in the ring R if there exists a subset S ⊂ R with L = { x ∈ R | xS = 0}. In this case we write L = l(S) . A right annihilator K = r(S) is defined similarly. Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings. Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

Locations

• Canadian Journal of Mathematics - View - PDF