Amini, Babak; Amini, Afshin; Facchini, Alberto Equivalence of diagonal matrices over local rings. (English) Zbl 1158.16011 J. Algebra 320, No. 3, 1288-1310 (2008). The article uses two equivalence relations of finitely presented modules: (1) epigeny, when two modules are equivalent if and only if either of them is an epimorphic image of the other; (2) lower part, which is a dual of epigeny in a certain sense. The main result (Theorem 5.3) is a Krull-Schmidt type theorem: two finite direct sums of non-zero cyclic modules over local rings are isomorphic if and only if both the epigeny classes and the lower parts of the summands of the sums are the same, counted with multiplicity. However, the pairing of epigeny classes and lower parts induced by the summands may differ in the two sums. The proof uses typical arguments over local rings, based on similar results on uniserial modules. A notable aspect is that the endomorphism ring of cyclic modules is semi-local with at most two maximal ideals (Theorem 2.1), which correspond to the epigeny class and lower part. The authors claim without proof that their results extend to locally presented modules and couniformly presented modules over semi-perfect rings. Reviewer: Gábor Braun (Budapest) Cited in 1 ReviewCited in 24 Documents MSC: 16L30 Noncommutative local and semilocal rings, perfect rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16S50 Endomorphism rings; matrix rings Keywords:isomorphism classes of direct sums of cyclically presented modules over local rings; Krull-Schmidt theorem; equivalences of diagonal matrices; direct sums of cyclic modules; endomorphism rings PDFBibTeX XMLCite \textit{B. Amini} et al., J. Algebra 320, No. 3, 1288--1310 (2008; Zbl 1158.16011) Full Text: DOI References: [1] Anderson, F. W.; Fuller, K. R., Rings and categories of modules, Grad. Texts in Math., vol. 13 (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0242.16025 [2] Diracca, L.; Facchini, A., Uniqueness of monogeny classes for uniform objects in abelian categories, J. Pure Appl. Algebra, 172, 183-191 (2002) · Zbl 1006.18010 [3] Facchini, A., Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc., 348, 4561-4575 (1996) · Zbl 0868.16003 [4] Facchini, A., Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progr. Math., vol. 167 (1998), Birkhäuser: Birkhäuser Basel · Zbl 0930.16001 [5] Facchini, A., Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids, J. Algebra, 256, 280-307 (2002) · Zbl 1016.16002 [6] Facchini, A.; Halter-Koch, F., Projective modules and divisor homomorphisms, J. Algebra Appl., 2, 435-449 (2003) · Zbl 1058.16002 [7] Facchini, A.; Herbera, D., Local morphisms and modules with a semilocal endomorphism ring, Algebr. Represent. Theory, 9, 403-422 (2006) · Zbl 1130.16014 [8] Facchini, A.; Příhoda, P., Monogeny dimension relative to a fixed uniform module, J. Pure Appl. Algebra (2008), in press · Zbl 1159.16003 [9] A. Facchini, P. PřÍhoda, Representations of the category of serial modules of finite Goldie dimension, in: R. Göbel, B. Goldsmith (Eds.), Contributions to Module Theory, de Gruyter, 2008, in press; A. Facchini, P. PřÍhoda, Representations of the category of serial modules of finite Goldie dimension, in: R. Göbel, B. Goldsmith (Eds.), Contributions to Module Theory, de Gruyter, 2008, in press · Zbl 1221.16009 [10] Fitting, H., Über den Zusammenhang zwischen dem Begriff der Gleichartigkeit zweier Ideale und dem Äquivalenzbegriff der Elementarteilertheorie, Math. Ann., 112, 572-582 (1936) · JFM 62.0091.04 [11] Kaplansky, I., Elementary divisors and modules, Trans. Amer. Math. Soc., 66, 464-491 (1949) · Zbl 0036.01903 [12] Levy, L.; Robson, J. C., Matrices and pairs of modules, J. Algebra, 29, 427-454 (1974) · Zbl 0282.16001 [13] Puninski, G., Some model theory over a nearly simple uniserial domain and decompositions of serial modules, J. Pure Appl. Algebra, 163, 3, 319-337 (2001) · Zbl 1025.16005 [14] Puninski, G., Projective modules over the endomorphism ring of a biuniform module, J. Pure Appl. Algebra, 188, 227-246 (2004) · Zbl 1055.16001 [15] Warfield, R. B., Decomposability of finitely presented modules, Proc. Amer. Math. Soc., 25, 167-172 (1970) · Zbl 0204.05902 [16] Warfield, R. B., Serial rings and finitely presented modules, J. Algebra, 37, 187-222 (1975) · Zbl 0319.16025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.