Abstract
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 \(\oplus _{i=1}^{n}M_{i}\), that is, block-diagonal decompositions, where each object Mi of Morph(Mod-R) denotes a morphism \(\mu _{M_{i}}\colon M_{0,i}\to M_{1,i}\) 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 \(\oplus _{i=1}^{n}M_{i}\) depend on four invariants.
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Presented by: Kenneth Goodearl
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The first and the third authors were partially supported by Dipartimento di Matematica “Tullio Levi-Civita” of Università di Padova (Project BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and Research program DOR1828909 “Anelli e categorie di moduli”).
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Campanini, F., El-Deken, S.F. & Facchini, A. Homomorphisms with Semilocal Endomorphism Rings Between Modules. Algebr Represent Theor 23, 2237–2256 (2020). https://doi.org/10.1007/s10468-019-09936-x
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DOI: https://doi.org/10.1007/s10468-019-09936-x