Abstract
In this paper, we show that modules with semilocal endomorphism rings appear in abundance in applications, that their direct-sum decompositions are described by the so-called Krull monoids, and that this implies a geometric regularity of the direct-sum decompositions of these modules. Their direct-sum decompositions into indecomposables are not necessarily unique in the sense of the Krull-Schmidt theorem. The application of the theory of Krull monoids to the study of direct-sum decompositions of modules has been developed during the last five years. After a quick survey of the results obtained in this direction, we concentrate in particular on the abundance of examples. At present, these examples are scattered in the literature, and we try to collect them in a systematic way.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. M. Bergman, “Coproducts and some universal ring constructions,” Trans. Amer. Math. Soc., 200, 33–88 (1974).
G. M. Bergman and W. Dicks, “Universal derivations and universal ring constructions,” Pacific J. Math., 79, 293–337 (1978).
G. Calugareanu, “Abelian groups with semi-local endomorphism ring,” Comm. Algebra, 30, No. 9, 4105–4111 (2002).
R. Camps and W. Dicks, “On semi-local rings,” Israel J. Math., 81, 203–211 (1993).
L. G. Chouinard II, “Krull semigroups and divisor class groups,” Can. J. Math., 33, 1459–1468 (1981).
A. Facchini, “Krull-Schmidt fails for serial modules,” Trans. Amer. Math. Soc., 348, 4561–4575 (1996).
A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Math., Vol. 167, Birkhäuser (1998).
A. Facchini, “Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids,” J. Algebra, 256, 280–307 (2002).
A. Facchini and F. Halter-Koch, “Projective modules and divisor homomorphisms,” J. Algebra Appl., 2, No. 4, 435–449 (2003).
A. Facchini and D. Herbera, “K 0 of a semilocal ring,” J. Algebra, 225, 47–69 (2000).
A. Facchini and D. Herbera, “Two results on modules whose endomorphism ring is semilocal,” Algebr. Represent. Theory, to appear, 2004.
A. Facchini and D. Herbera, “Local morphisms and modules with a semilocal endomorphism ring,” to appear, 2004.
A. Facchini and R. Wiegand, “Direct-sum decompositions of modules with semilocal endomorphism ring,” J. Algebra, 274, 689–707 (2004).
D. Herbera and A. Shamsuddin, “Modules with semi-local endomorphism ring,” Proc. Amer. Math. Soc., 123, 3593–3600 (1995).
E. L. Lady, “Summands of finite rank torsion-free Abelian groups,” J. Algebra, 32, 51–52 (1974).
P. Prihoda, “Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension,” J. Algebra, to appear, 2004.
W. Vasconcelos, “On finitely generated flat modules,” Trans. Amer. Math. Soc., 138, 505–512 (1969).
R. B. Warfield, Jr., “Serial rings and finitely presented modules,” J. Algebra, 37, 187–222 (1975).
R. B. Warfield, Jr., “Cancellation of modules and groups and stable range of endomorphism rings,” Pacific J. Math., 91, 457–485 (1980).
R. Wiegand, “Direct-sum decompositions over local rings,” J. Algebra, 240, 83–97 (2001).
Author information
Authors and Affiliations
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 231–244, 2004.
Rights and permissions
About this article
Cite this article
Facchini, A. Geometric regularity of direct-sum decompositions in some classes of modules. J Math Sci 139, 6814–6822 (2006). https://doi.org/10.1007/s10958-006-0393-2
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0393-2