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Multiplication modules relative to torsion theories. (English) Zbl 0849.13008

Let \(R\) be a commutative unitary ring. An \(R\)-module \(M\) is called a multiplication module if for every submodule \(N \geq M\) there exists an ideal \(I \leq R\) such that \(N = M.I\). The authors relativize the above definition with respect to a torsion theory on \(R\). The starting point of the theory is the following examples: While Dedekind domains are multiplication rings, Krull domains are multiplication rings with respect to the torsion theory corresponding to the canonical topology. Some general properties of relative multiplication modules are studied, as well as some local properties.

MSC:

13D30 Torsion theory for commutative rings
13C12 Torsion modules and ideals in commutative rings
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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