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On secondary modules over pullback rings. (English) Zbl 1011.13004

The pullback \(R\) of two local Dedekind domains \(R_1, R_2\) mapped into a common field is called the pullback ring; it is a commutative Noetherian local ring with unique maximal ideal \(P\). An \(R\)-module \(S\) is said to be separated, if there are \(R_i\)-modules \(S_i\), such that \(S\) is a submodule of \(S_1\oplus S_2\). A non-zero \(R\)-module is said to be secondary, if every multiplication by \(r\in R\) is either surjective or nilpotent. In this case \(J= \text{Rad(Ann}_RM)\) is the prime radical and \(M\) is said to be \(J\)-secondary. If \(M=M_1+\dots+M_n\) is a minimal secondary representation of \(M\), where \(M_i\) are \(J_i\)-secondary, then the set of prime ideals attached to \(M\) is \(\{ J_1,\dots,J_n\}\). If such a representation exists, then \(M\) is said to be representable.
This paper is a continuation of the author’s preceding papers [S. Ebrahimi Atani, Commun. Algebra 28, No. 9, 4037-4069 (2000; Zbl 0960.16002)]; it classifies indecomposable representable modules with finite dimensional top \(M/\text{Rad}(R)M\) over \(R\).
First all indecomposable secondary separated \(R\) modules are given, then, with the aid of this list it is shown that non-separated indecomposable representable \(R\)-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable secondary modules. One of the results is as follows:
Let \(R\) be the pullback ring of two discrete valuation domains with common factor field. Then the indecomposable non-separated representable modules with finite-dimensional top are as follows:
(1) The indecomposable modulesof finite length (except the separated \(R/P\)),
(2) the so-called doubly infinite representable modules,
(3) the so-called singly infinite representable modules, except two specific Prüfer modules.
One consequence is that every indecomposable representable non-separated \(R\)-module with finite-dimensional top is pure injective.

MSC:

13C11 Injective and flat modules and ideals in commutative rings
13D02 Syzygies, resolutions, complexes and commutative rings

Citations:

Zbl 0960.16002
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References:

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