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DIRECT SUMS OF INFINITELY MANY KERNELS

Part of: Modules, bimodules and ideals Local rings and generalizations

Published online by Cambridge University Press:  23 November 2010

ŞULE ECEVIT ,
ALBERTO FACCHINI  and
M. TAMER KOŞAN
ŞULE ECEVIT
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey (email: [email protected])
ALBERTO FACCHINI*
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, 35121 Padova, Italy (email: [email protected])
M. TAMER KOŞAN
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let 𝒦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

Keywords

moduledirect sum decompositionsemilocal endomorphism ring

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16L30: Noncommutative local and semilocal rings, perfect rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 2 , October 2010 , pp. 199 - 214
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

Alberto Facchini was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (Prin 2007 ‘Rings, algebras, modules and categories’) and by the Università di Padova (Progetto di Ricerca di Ateneo CPDA071244/07).

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