Finitistic dimension and a homological generalization of semi-primary rings

TL;DR: In this article, Kaplansky showed that a commutative ring R is left T-nilpotent if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0.

Abstract: Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left \"global\" dimensions of R. Auslander and Buchsbaum [3] and, subsequently, Serre [14], found it relevant and fruitful, in the study of commutative Noetherian rings, to introduce a \"finitistic global\" dimension defined by restricting the supremum of projective dimensions to finitely generated modules of finite projective dimension. The impressive theory developed by these authors prompted Kaplansky to consider, for general commutative rings, a similar finitistic dimension (waiving the restriction, in the above, to finitely generated modules), and, in a seminar at the University of Chicago (1958), he proved the theorem below, characterizing commutative rings R for which this dimension vanishes. This result is the origin of the present paper. Definition. If N is an ideal in a ring P, we say that N is left T-nilpotent i\"T\" ior transfinite) if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0. iRight T-nilpotence requires instead that a„ • • • oi = 0.)