Generalizations of Fitting's Lemma in arbitrary associative rings
by
Michael P. Drazin
Department of Mathematics, Purdue University

Fitting's lemma, while usually stated for all endomorphisms a of a module V (i.e. a in End V) with chain conditions, is easily translatable so as to apply to (suitable) elements a of an arbitrary associative ring R in terms of "core-nilpotent" decompositions a = c + n (cf. Greville 1967); it is also easy to obtain necessary and sufficient conditions on a for the existence and uniqueness of such decompositions, in terms of strong \pi-regularity (which is equivalent to having a pseudo-inverse in the sense of the speaker (1958)).

The main results concern weaker "core-quasi-nilpotent" decompositions a = c + n for a larger class of a in R, where existence and (to some extent) uniqueness can be characterized in terms of another more general generalized inverse and an appropriate natural partial order.

Date received: March 11, 1999