Homomorphisms with Semilocal Endomorphism Rings Between Modules

Campanini, Federico;El-Deken, Susan F.;Facchini, Alberto
(2019) Algebras and Representation Theory — Vol. 23, n° 6, p. 2237-2256 (2019)

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Abstract
(en) We study the category Morph(Mod-R) whose objects are all morphisms between two right R-modules. The behavior of the objects of Mod-R whose endomorphism ring in Morph(Mod-R) is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum , that is, block-diagonal decompositions, where each object Mi of Morph(Mod-R) denotes a morphism and where all the modules Mj,i have a local endomorphism ring End(Mj,i), depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules Mj,i are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum depend on four invariants.
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  • University of Padova

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Campanini, F., El-Deken, S. F., & Facchini, A. (2019). Homomorphisms with Semilocal Endomorphism Rings Between Modules. Algebras and Representation Theory, 23(6), 2237-2256. https://doi.org/10.1007/s10468-019-09936-x (Original work published 2019)