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Idealization and theorems of D. D. Anderson. II. (English) Zbl 1136.13007

This paper is the second in a sequence [the first is Commun. Algebra 34, 4479–4501 (2006; Zbl 1109.13010)] by the author using the method of idealization to study multiplication modules, cancellation modules, and related modules over a commutative ring \(R\). Submodules \(N\) of a faithful multiplication module \(M\) characterized by properties like \(\bigcap I_{i}M=(\bigcap I_{i})M\) for every collection \(\{I_{i}\}\) of ideals of \(R\) are studied. Particular attention is given to the question of when a homogeneous ideal \(I(+)N\) of the idealization \(R(M)\)of the \(R\)-module \(M\) has certain properties such as being free, projective, flat, dense, or multiplication. In many cases this forces \(N=IM\). Finally the author considers when \(R(M)\)is a Prüfer ring, Bezout ring, valuation ring, additively regular ring, or Marot ring, has property (A), satisfies (a.c.), or is a GCDring or a GGCD ring, usually in the case where \(M\) is a torsion-free \(R\)-module.

MSC:

13C13 Other special types of modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings

Citations:

Zbl 1109.13010
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References:

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