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A
counter-example to a lemma of Skornjakov
Barbara Osofsky |
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Vol. 15 (1965), No. 3, 985–987
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Abstract |
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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
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Milestones
Received: 5 August 1964
Revised: 18 March 1965
Published: 1 September 1965
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Authors |
| Barbara Osofsky | |
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