Homomorphisms with Semilocal Endomorphism Rings Between Modules
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Homomorphisms with Semilocal Endomorphism Rings Between Modules Federico Campanini1 · Susan F. El-Deken2 · Alberto Facchini1 Received: 18 September 2018 / Accepted: 12 November 2019 / © Springer Nature B.V. 2019
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 ⊕ni=1 Mi , that is, block-diagonal decompositions, where each object Mi of Morph(Mod-R) denotes a morphism μMi : M0,i → M1,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 directsum 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 ⊕ni=1 Mi depend on four invariants. Keywords Module morphism · Semilocal ring · Direct-sum decomposition Mathematics Subject Classification (2010) Primary 16D70 · 16L30 · 16S50
Presented by: Kenneth Goodearl The first and the third authors were partially supported by Dipartimento di Matematica “Tullio Levi-Civita” of Universit`a di Padova (Project BIRD163492/16 “Categorical homological methods in the study of algebraic structures” and Research program DOR1828909 “Anelli e categorie di moduli”). Federico Campanini
[email protected] Susan F. El-Deken [email protected] Alberto Facchini [email protected] 1
Dipartimento di Matematica, Universit`a di Padova, 35121 Padova, Italy
2
Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, 11790, Helwan, Cairo, Egypt
F. Campanini et al.
1 Introduction The study of block decompositions of matrices is one of the classical themes in Linear Algebra. We refer to the description of matrices up to the matrix equivalence ∼ defined, for any two rectangular m × n matrices A and B, by A ∼ B if B = Q−1 AP for some invertible n × n matrix P and some invertible m × m matrix Q. Recently, the case of matrices with entries in an arbitrary local ring has sparked interest [16]. In [1, Corollary 5.4], B. Amini, A. Amini and A. Facchini considered the case of diagonal matrices over local rings, proving that the equivalence of two such matrices depends on two invariants, called lower part and epigeny class. That is, if a1 , . . . , an , b1 , . . . , bn are elements of a local ring R, then diag(a1 , . . . , an ) ∼ diag(b1 , . . . , bn ) if and only if there are two permutations σ, τ of {1, 2, . . . , n} such that the cyclically presented right R-modules R/ai R and R/bσ (i) R have the same lower part and R/ai R, R/bτ (i) R have the same epigeny class, for every i = 1, 2, . . . , n. Thus the block decomposition of a matrix with entries in a ring is not unique, that is, the blocks on two equ
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