Endomorphism rings of reduced complete torsion-free modules over complete discrete valuation rings

Abstract

The purpose of this paper is to give necessary and sufficient conditions for an abstract ring to be isomorphic to the endomorphism ring of a reduced complete torsion-free module over a complete discrete valuation ring. In [3] we have characterized the endomorphism rings of reduced complete torsion-free modules over (not necessarily commutative) complete discrete valuation rings. By generalizing the Harrison-Matlis duality of [1] and [4] to the noncommutative case, this work simultaneously characterizes endomorphism rings of divisible torsion modules. The model for our main result was Wolfson's beautiful characterization of the ring of all linear transformations of a vector space over a division ring in [5]. The purpose of this note is to show how Wolfson's theorem can be used directly for the characterization of the endomorphism rings of these modules. WOLFSON'S THEOREM. Let E be a ring and Eo its right socle. Then the followving are equivalent: I. E is isomorphic to the ring of all linear transformations of a vector space over a division ring. II. (1) Eo is not a zero-ring, and is contained in every nonzero twvo-sided ideal of E. (2) If L is a left ideal of E w hich is annihilated on the right only by zero, then EOczL. (3) The sum of twvo left (right) annihilators is a left (right) annihilator. (4) E possesses an identity element. DEFINITION. A ring which satisfies condition I or II in the theorem above is said to be a Wolfson ring. Received by the editors February 28, 1972. AMS (MOS) subject classi{/cations (1970). Primary 16A48, 16A42, 16A64; Secondary 16A80, 20K20.