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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Lattices of normally indecomposable modules


Author: Juliusz Brzezinski
Journal: Proc. Amer. Math. Soc. 68 (1978), 271-276
MSC: Primary 16A64; Secondary 16A46, 18E05
MathSciNet review: 0469979
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Abstract: If M, N are finitely generated left R-modules, then M divides $ {N^1}$ if there is an epimorphism $ {M^{(r)}} \to N$. M is normally indecomposable if $ M \cong {M_1} \oplus {M_2}$ and $ {M_1}$ divides $ {M_2}$ imply $ {M_2} = 0$. If R is an Artin algebra or an order over a complete discrete valuation ring in a semisimple algebra, the set of isomorphism classes of normally indecomposable R-modules (respectively R-lattices) is partially ordered by the divisibility relation. We show that for R of finite representation type this partially ordered set is a lattice satisfying the Jordan-Dedekind chain condition and the length of maximal chains is equal to the number of isomorphism classes of indecomposable R-modules (respectively R-lattices).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0469979-4
PII: S 0002-9939(1978)0469979-4
Keywords: Krull-Schmidt theorem, normal decomposition of modules, Jordan-Dedekind chain condition, finite representation type
Article copyright: © Copyright 1978 American Mathematical Society