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The Krull-Schmidt Theorem in the Case Two

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Abstract

We study what happens if, in the Krull-Schmidt Theorem, instead of considering modules whose endomorphism rings have one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals. If a ring has exactly two maximal right ideals, then the two maximal right ideals are necessarily two-sided. We call such a ring of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs. The vertices of the graphs are ideals in a suitable full subcategory of Mod-R. The edges are isomorphism classes of modules. The complete bipartite graphs give rise to a behavior described by a Weak Krull-Schmidt Theorem. Such a behavior had been previously studied for the classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, and couniformly presented modules. All these modules have endomorphism rings that are either local or of type 2. Here we present a general theory that includes all these cases.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Università di Padova, 35121, Padova, Italy

    Alberto Facchini

  2. Faculty of Mathematics and Physics, Department of Algebra, Charles University in Prague, Sokolovská 83, 186 75, Prague, Czech Republic

    Pavel Příhoda

Authors
  1. Alberto Facchini
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  2. Pavel Příhoda
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Correspondence to Alberto Facchini.

Additional information

Alberto Facchini was partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca, Italy (Prin 2007 “Rings, algebras, modules and categories”) and by Università di Padova (Progetto di Ricerca di Ateneo CPDA071244/07).

Pavel Příhoda was partially supported by Research Project MSM 0021620839 and grant GACR 201/09/0816.

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Facchini, A., Příhoda, P. The Krull-Schmidt Theorem in the Case Two. Algebr Represent Theor 14, 545–570 (2011). https://doi.org/10.1007/s10468-009-9202-1

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