Vol. 15, No. 3, 1965

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A counter-example to a lemma of Skornjakov

Barbara Osofsky

Vol. 15 (1965), No. 3, 985–987
Abstract

In his paper, Rings with injective cyclic modules, translated in Soviet Mathematics 4 (1963), p. 36–39, L. A. Skornjakov states the following lemma: If a cyclic R-module M and all its cyclic submodules are injective, then the partially ordered set of cyclic submodules of M is a complete, complemented lattice.

An example is constructed to show that this lemma is false, thus invalidating Skorniakov’s proof of the theorem: Let R be a ring all of whose cyclic modules are injective. Then R is semi-simple Artin. The theorem, however, is true. (See Osofsky [4].)

Mathematical Subject Classification
Primary: 16.40
Secondary: 18.00
Milestones
Received: 5 August 1964
Revised: 18 March 1965
Published: 1 September 1965
Authors
Barbara Osofsky