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Krull-Schmidt fails for serial modules. (English) Zbl 0868.16003
In 1995, D. Herbera, L. S. Levy, P. Vamos and the author [Proc. Am. Math. Soc. 123, No. 12, 3587-3592 (1995; Zbl 0847.16013)] showed that the Krull-Schmidt Theorem fails for artinian modules, thus answering a 1932 question of Krull. In 1975, R. B. Warfield [J. Algebra 37, 187-222 (1975; Zbl 0319.16025)] proved that every finitely presented module over a serial ring is a direct sum of uniserial modules and asked about the uniqueness of these decompositions. (In 1949, I. Kaplansky [Trans. Am. Math. Soc. 66, 464-491 (1949; Zbl 0036.01903)] showed that the summands are unique if the ring is commutative.) The author shows that for every positive integer \(n\) there exist a serial ring \(R\) and \(2n\) pairwise non-isomorphic finitely presented uniserial \(R\)-modules \(U_i\), \(V_i\) (\(1\leq i\leq n\)) such that \(U_1\oplus\dots\oplus U_n\cong V_1\oplus\dots\oplus V_n\). It is also proved that the Grothendieck group of the class of serial modules of finite Goldie dimension is free abelian and that the endomorphism rings of such modules are semilocal.

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S50 Endomorphism rings; matrix rings
16D90 Module categories in associative algebras
Full Text: DOI
[1] Rosa Camps and Warren Dicks, On semilocal rings, Israel J. Math. 81 (1993), no. 1-2, 203 – 211. · Zbl 0802.16010
[2] Alberto Facchini, Dolors Herbera, Lawrence S. Levy, and Peter Vámos, Krull-Schmidt fails for Artinian modules, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3587 – 3592. · Zbl 0847.16013
[3] Alberto Facchini and Luigi Salce, Uniserial modules: sums and isomorphisms of subquotients, Comm. Algebra 18 (1990), no. 2, 499 – 517. · Zbl 0712.16008
[4] Dolors Herbera and Ahmad Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3593 – 3600. · Zbl 0843.16017
[5] I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. · Zbl 0036.01903
[6] B. Stenström, Rings of Quotients, Springer-Verlag, Berlin, 1975. · Zbl 0296.16001
[7] K. Varadarajan, Dual Goldie dimension, Comm. Algebra 7 (1979), no. 6, 565 – 610. · Zbl 0487.16019
[8] R. B. Warfield Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), no. 2, 187 – 222. · Zbl 0319.16025
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