Abd El-Bast, Zeinab; Smith, Patrick F. Multiplication modules. (English) Zbl 0642.13002 Commun. Algebra 16, No. 4, 755-779 (1988). 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 Cited in 6 ReviewsCited in 156 Documents 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 Keywords:multiplication module; finiteness conditions Citations:Zbl 0468.13011 PDF BibTeX XML Cite \textit{Z. Abd El-Bast} and \textit{P. F. Smith}, Commun. Algebra 16, No. 4, 755--779 (1988; Zbl 0642.13002) Full Text: DOI OpenURL 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 [5] Chatters A.W., Rings with chain conditions (1980) · Zbl 0446.16001 [6] Gordon R., Krull dimension 133 (1973) [7] Kaplansky I., commutative rings (1970) [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.