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Every ring in this note has an identity and every module is a unitary left module. If R is a ring then an .R-module M is said to be distributive if the lattice of submodules of M is distributive. The purpose of this note is to establish the result in the title i.e. to prove the following: PROPOSITION. A finitely generated, Artinian and distributive module is cyclic. Two reasons are offered for the publication of this elementary result. Firstly, I believe that it is interesting in its own right but seems to have been missed by writers on distributive modules: [1, Chap. IX], [2], [3, Chap. 4], [7]. The second reason is Fuller's work [5], [6] on rings of distributive module type. These are rings for which every module is a direct sum of distributive modules. Using sophisticated methods one can show [5, Remark 7(2)] that these are rings of finite representation type and every module is in fact a direct sum of distributive modules of finite length. Our result immediately implies that the modules over a ring of distributive module type are direct sums of cyclic modules. Rings with this latter property have received a great deal of attention.
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It is typically read by researchers, students, and practitioners in Mathematics.
- Author
- P. Vámos
- Publisher
- Wiley
- Published
- 1978
- Language
- EN
- Field
- Mathematics (Physical Sciences)