Dedekind finite objects in module categories

AbstractAn object A in a category is called Dedekind finite if every monomorphism f : A→A is an isomorphism. This notion generalizes a characterizing property of finite sets and finite- dimensional vector spaces. We show that a free module of finite rank over a commutative ring R is Dedekind finite if and only if R is a quoring (i.e., every regular element of R is invertible). We also describe completely the divisible R-modules that are Dedekind finite, R a Dedekind domain