# Some characterizations of regular modules.

• Volume: 34, Issue: 2, page 241-248
• ISSN: 0214-1493

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## Abstract

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Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M is locally split.(3) M is locally projective and every cyclic submodule of M is a direct summand of M.

## How to cite

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Azumaya, Goro. "Some characterizations of regular modules.." Publicacions Matemàtiques 34.2 (1990): 241-248. <http://eudml.org/doc/41129>.

@article{Azumaya1990,
abstract = {Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M is locally split.(3) M is locally projective and every cyclic submodule of M is a direct summand of M.},
author = {Azumaya, Goro},
journal = {Publicacions Matemàtiques},
keywords = {Teoría de anillos; Anillos; Módulos; regular module; locally projective module; flat strict Mittag-Leffler module; locally split homomorphism; characterizations of regularity; left R-module; Zelmanowitz-regular; cyclic submodule; direct summand},
language = {eng},
number = {2},
pages = {241-248},
title = {Some characterizations of regular modules.},
url = {http://eudml.org/doc/41129},
volume = {34},
year = {1990},
}

TY - JOUR
AU - Azumaya, Goro
TI - Some characterizations of regular modules.
JO - Publicacions Matemàtiques
PY - 1990
VL - 34
IS - 2
SP - 241
EP - 248
AB - Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M is locally split.(3) M is locally projective and every cyclic submodule of M is a direct summand of M.
LA - eng
KW - Teoría de anillos; Anillos; Módulos; regular module; locally projective module; flat strict Mittag-Leffler module; locally split homomorphism; characterizations of regularity; left R-module; Zelmanowitz-regular; cyclic submodule; direct summand
UR - http://eudml.org/doc/41129
ER -

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