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.
Similar content being viewed by others
References
Amini, B., Amini, A., Facchini, A.: Equivalence of diagonal matrices over local rings. J. Algebra 320, 1288–1310 (2008)
Amini, A., Amini, B., Facchini, A.: Weak Krull-Schmidt for infinite direct sums of cyclically presented modules over local rings. Rend. Semin. Mat. Univ. Padova (2009, in press)
Anderson, F.W., Fuller, K.R.: Rings and categories of modules, second edition. In: GTM, vol. 13. Springer, New York (1992)
Bican, L.: Weak Krull-Schmidt Theorem. Comment. Math. Univ. Carol. 39, 633–643 (1998)
Calugareanu, G.: Abelian groups with semi-local endomorphism ring. Commun. Algebra 30(9), 4105–4111 (2002)
Camps, R., Dicks, W.: On semilocal rings. Isr. J. Math. 81, 203–211 (1993)
Corner, A.L.S.: Every countable reduced torsion-free ring is an endomorphism ring. Proc. Lond. Math. Soc. 13(3), 687–710 (1963)
Facchini, A.: Krull-Schmidt fails for serial modules. Trans. Am. Math. Soc. 348, 4561–4575 (1996)
Facchini, A.: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. In: Progress in Math., vol. 167. Birkhäuser, Basel (1998)
Facchini, A.: Injective modules, spectral categories, and applications. In: Jain, S.K., Parvathi, S. (eds.) Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications. Contemporary Math, vol. 456, pp. 1–17. American Mathematical Society, Providence (2008)
Facchini, A., Girardi, N.: Couniformly presented modules and dualities. In: Van Huynh, D., López Permouth, S.R. (eds.) Advances in Ring Theory. Trends in Math. Birkhäuser, Basel (2010)
Facchini, A., Herbera, D.: Two results on modules whose endomorphism ring is semilocal. Algebr. Represent. Theory 7(5), 575–585 (2004)
Facchini, A., Herbera, D., Levy, L.S., Vámos, P.: Krull-Schmidt fails for Artinian modules. Proc. Am. Math. Soc. 123, 3587–3592 (1995)
Facchini, A., Ecevit, Ş., Koşan, M.T., Özdin, T.: Kernels of morphisms between indecomposable injective modules. Glasgow Math. J. (2010, in press)
Facchini, A., Příhoda, P.: Factor categories and infinite direct sums. Int. Electron. J. Algebra 5, 135–168 (2009)
Facchini, A., Příhoda, P.: Endomorphism rings with finitely many maximal right ideals. Commun. Algebra (2010, in press)
Facchini, A., Wiegand, R.: Direct-sum decompositions of modules with semilocal endomorphism ring. J. Algebra 274, 689–707 (2004)
Gabriel, P., Oberst, U.: Spektralkategorien und reguläre Ringe im von-Neumannschen Sinn. Math. Z. 92, 389–395 (1966)
Göbel, R., Trlifaj, J.: Cotilting and a hierarchy of almost cotorsion groups. J. Algebra 224, 110–122 (2000)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-009-9202-1