HIGH POWERS IN ENDOMORPHISM RINGS OVER DEDEKIND DOMAINS

Let A be a Dedekind domain and T an endomorphism of a finitely generated projective A-module. If T is an s-th power in End_A(M) for s ranging over an infinite set S of positive integers, then (a) T decomposes as a direct sum of the zero operator and an invertible operator on a summand of M and (b) that summand is semisimple or of finite order if S is appropriately large (what this means depends on the structure of the additive and multiplicative groups of A). This generalizes a result of Cavachi’s to the effect that the only nonsingular integer matrix that is an s-th power in M_n(Z) for all s is the identity.