Dedekind-infinite Set - Generalizations

Generalizations

Expressed in category-theoretical terms, a set A is Dedekind-finite if in the category of sets, every monomorphism f: A → A is an isomorphism. A von Neumann regular ring R has the analogous property in the category of (left or right) R-modules if and only if in R, xy = 1 implies yx = 1. More generally, a Dedekind-finite ring is any ring that satisfies the latter condition. Beware that a ring may be Dedekind-finite even if its underlying set is Dedekind-infinite, e.g. the integers.