Finitely Generated Modules over Pullback Rings

Abstract

The purpose of this paper is to outline a new approach to the classification of finitely generated indecomposable modules over certain kinds of pullback rings. If R is the pullback of two hereditary noetherian homogeneously serial rings, finitely generated over their centers, over a common semi-simple artinian ring, then this classification can be divided into the classification of indecomposable artinian modules and those modules over the coordinate rings with no non-trivial artinian submodules. The classification of the artinian modules can be reduced to the case of a finite dimensional algebra over a semi-simple ring. This approach is then used as the key step in the case where the coordinate rings are hereditary noetherian serial rings over a common quotient which is a matrix ring over a field, resulting in a complete module classification.

Absorbing Multiplication Modules Over Pullback Rings

“…We continue to use the notation already established, so R is the pullback ring as in (1). In this section we find the indecomposable non-separated 2-absorbing multiplication modules with finite-dimensional top.…”

“…Lemma 3.3. Let R be the pullback ring as in (1). Then, up to isomorphism, the following separated R-modules are indecomposable and 2-absorbing multiplication: For each i, let E i be the R i -injective hull of R i /P i , regarded as an R-module, so E 1 , E 2 are the modules listed under (ii) in Lemma 3.3.…”

“…Let S = R be an indecomposable separated 2-absorbing multiplication module over the pullback ring as in (1). Then S is isomorphic to one of the modules listed in Lemma 3.3.…”

“…Furthermore, for i = j, 0 → P i → R → R j → 0 is an exact sequence of R-modules (see [20]). Modules over pullback rings has been studied by several authors (see for example, [1,5,9,15,19,25,32]). …”

Computing the Additive Structure of Indecomposable Modules Over Dedekind-Like Rings Using Gröbner Bases

“…In 1949, Szekeres started the classification and matrix description of modules over Z p n C p . Since then it has been studied in detail, see [1,3,9,10,11,12,14]. In [9,10], Levy studied these modules in the more general context of modules over a pullback of two Dedekind rings with a common field, which he called Dedekind-like rings.…”