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Multiplication modules. (English) Zbl 0642.13002

An in, e.g. the paper by A. Barnard in J. Algebra 71, 174-178 (1981; Zbl 0468.13011), M a multiplication module over a commutative ring R with identity means here that every submodule has the form IM for some ideal I of R. Various characterisations of multiplication modules are given. If R has few zero-divisors, faithful multiplication modules are isomorphic to invertible ideals. Let M be a multiplication module. In many ways the structure of M reflects that of R. Thus submodules lie in maximal submodules and the latter have the form PM where P is a uniquely determined maximal ideal. Much attention is paid to finiteness conditions: M Artinian, Noetherian etc., e.g. if the set of minimal prime submodules of M is finite then M is f.g. If M is f.g., the lattice of ideals of R and the lattice of R-submodules of M are isomorphic.
Reviewer: C.P.L.Rhodes

MSC:

13A05 Divisibility and factorizations in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13E05 Commutative Noetherian rings and modules
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13C05 Structure, classification theorems for modules and ideals in commutative rings

Citations:

Zbl 0468.13011
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References:

[1] DOI: 10.4153/CJM-1976-072-1 · Zbl 0343.13009
[2] Anderson D.D., Math. Japonica 4 pp 463– (1980)
[3] Anderson F.W., Rings and categories of modules (1974) · Zbl 0301.16001
[4] DOI: 10.1016/0021-8693(81)90112-5 · Zbl 0468.13011
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[8] DOI: 10.4153/CMB-1986-006-7 · Zbl 0546.13001
[9] Mott, J.L. 1969.Multiplication rings containing only finitely many minimal prime ideals, ser. A-l Vol. 33, 73–83. J. Sci. Hiroshima univ. · Zbl 0184.29202
[10] DOI: 10.4153/CMB-1979-013-9 · Zbl 0408.13002
[11] DOI: 10.4153/CJM-1969-116-7 · Zbl 0183.04001
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