The Jacobson Radical’s Role in Isomorphism Theorems for p-Adic Modules Extends to Topological Isomorphism

Abstract

For a complete discrete valuation domain R, a class of R-modules is said to satisfy an isomorphism theorem if an isomorphism between the endomorphism algebras of two modules in that class implies that the modules are isomorphic. A class satisfies a Jacobson radical isomorphism theorem if an isomorphism between only the Jacobson radicals of the endomorphism rings of two modules in that class implies that the modules are isomorphic. Jacobson radical isomorphism theorems exist for subclasses of the classes of torsion, torsion-free and mixed modules which satisfy an isomorphism theorem. Warren May investigated the use of the finite topology in isomorphism theorems, and showed that the topological setting allows an isomorphism theorem for a broader class of reduced mixed modules than the algebraic isomorphism alone. The purpose of this paper is to prove that the parallels that exist between isomorphism theorems and Jacobson radical isomorphism theorems extend to the topological setting. The main result is that the class of reduced modules over a complete discrete valuation domain which contain an unbounded torsion submodule and are divisible modulo torsion satisfy a topological Jacobson radical isomorphism theorem.