Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 3, page 903-919
  • ISSN: 0011-4642

Abstract

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Let R ⋉ M be a trivial extension of a ring R by an R - R -bimodule M such that M R , R M , ( R , 0 ) R ⋉ M and R ⋉ M ( R , 0 ) have finite flat dimensions. We prove that ( X , α ) is a Ding projective left R ⋉ M -module if and only if the sequence M ⊗ R M ⊗ R X ⟶ M ⊗ α M ⊗ R X → α X is exact and coker ( α ) is a Ding projective left R -module. Analogously, we explicitly describe Ding injective R ⋉ M -modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

How to cite

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Mao, Lixin. "Ding projective and Ding injective modules over trivial ring extensions." Czechoslovak Mathematical Journal 73.3 (2023): 903-919. <http://eudml.org/doc/299126>.

@article{Mao2023,
abstract = {Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M_\{R\}$, $_\{R\}M$, $(R,0)_\{R\ltimes M\}$ and $_\{R\ltimes M\}(R,0)$ have finite flat dimensions. We prove that $(X,\alpha )$ is a Ding projective left $R\ltimes M$-module if and only if the sequence $M\otimes _R M\otimes _R X\stackrel\{M\otimes \alpha \}\{\longrightarrow \}M\otimes _R X\stackrel\{\alpha \}\{\rightarrow \}X$ is exact and $\{\rm coker\}(\alpha )$ is a Ding projective left $R$-module. Analogously, we explicitly describe Ding injective $R\ltimes M$-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.},
author = {Mao, Lixin},
journal = {Czechoslovak Mathematical Journal},
keywords = {trivial extension; Ding projective module; Ding injective module},
language = {eng},
number = {3},
pages = {903-919},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ding projective and Ding injective modules over trivial ring extensions},
url = {http://eudml.org/doc/299126},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Mao, Lixin
TI - Ding projective and Ding injective modules over trivial ring extensions
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 903
EP - 919
AB - Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M_{R}$, $_{R}M$, $(R,0)_{R\ltimes M}$ and $_{R\ltimes M}(R,0)$ have finite flat dimensions. We prove that $(X,\alpha )$ is a Ding projective left $R\ltimes M$-module if and only if the sequence $M\otimes _R M\otimes _R X\stackrel{M\otimes \alpha }{\longrightarrow }M\otimes _R X\stackrel{\alpha }{\rightarrow }X$ is exact and ${\rm coker}(\alpha )$ is a Ding projective left $R$-module. Analogously, we explicitly describe Ding injective $R\ltimes M$-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.
LA - eng
KW - trivial extension; Ding projective module; Ding injective module
UR - http://eudml.org/doc/299126
ER -

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