Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T05:46:59.318Z Has data issue: false hasContentIssue false
  • English
  • Français

Nil Subrings of Goldie Rings are Nilpotent

Published online by Cambridge University Press:  20 November 2018

Charles Lanski
Charles Lanski*
Affiliation:
University of Chicago, Chicago, Illinois
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. Lis 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 904 - 907
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Herstein, I. N. and Small, Lance, Nil rings satisfying certain chain conditions, Can. J. Math. 16 (1964), 771–776.Google Scholar