Cancellation and stability properties of generalized torsion modules

Abstract

Given a module $X$ over a ring $\mathrm{\Lambda }$ its stability class consists of all modules $X^{\prime }$ such that $X\oplus {\mathrm{\Lambda }}^a\cong X^{\prime }\oplus {\mathrm{\Lambda }}^b$ for some positive integers $a$, $b$. If the ring $\mathrm{\Lambda }$ is weakly finite then the stability class of a finitely generated $\mathrm{\Lambda }$-module has the structure of a tree. We show that if, in addition, $X$ is a generalized torsion module its stability class has the same shape as that of the zero module. In consequence we construct examples of nonprojective modules whose stability classes have arbitrarily large amounts of branching.