This paper is concerned with
deriving conditions which ensure that even though a module A may not necessarily
cancel from a direct sum A⊕B≅A⊕C, it can at least be concluded that Bn≅Cn for
some positive integer n. This conclusion is obtained from a type of stable range
condition on the endomorphism ring of A, which holds, for example, when A is a
finitely generated module over any subring of a finite-dimensional Q-algebra. As an
application of these methods to groups, it is shown that if A is a torsion-free abelian
group of finite rank, and B, C are arbitrary groups (not necessarily abelian) such
that A × B≅A × C, then there exists a positive integer n such that the direct
product of n copies of B is isomorphic to the direct product of n copies of
C.
