Some remarks on projective-free rings

Abstract

The homological classification of rings is based on the length of projective resolutions of modules, but it tells us nothing about the projective modules themselves. For information on them we have to turn to the monoid of projectives; thus for any ring R we have P(R), the monoid of (isomorphism classes of) finitely generated projective R-modules, with the operation [P ] + [Q] = [P ⊕ Q]. The simplest form that this monoid can take is P(R) ∼= N, the natural numbers, with [R] corresponding to 1. It means that every finitely generated projective R-module is free, of unique rank. Such a ring is said to be projective-free; in particular, since the rank of any free module is unique, such a ring has invariant basis number (IBN). Our aim in this note is to derive some results on projective-free rings. We shall show in Section 2 that surjective homomorphisms which are local, i.e. map non-units to non-units, do not diminish the monoid of projectives. In particular, we shall obtain conditions ensuring that a ring homomorphism reflects projective-freeness. A slightly weaker conclusion is reached in Section 3 by assuming the regularity of full matrices. I am indebted to G. M. Bergman whose comments on an earlier version greatly improved the presentation; in particular, Lemma 1 and suggestions leading to Theorem 2 are due to him. Throughout all rings are associative, with a unit element 1 which is preserved by homomorphisms and inherited by subrings.