A BAER-KAPLANSKY THEOREM FOR MODULES OVER PRINCIPAL IDEAL DOMAINS

摘要

We will prove that if G and H are modules over a principal ideal domain R such that the endomorphism rings End(R) (R circle plus G) and End(R) (R circle plus H) are isomorphic, then G congruent to H. Conversely, if R is a Dedekind domain such that two R-modules G and H are isomorphic whenever the rings End(R) (R circle plus G) and End(R) (R circle plus H) are isomorphic, then R is a PID.