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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.

MSC:
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
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[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
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