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Ding projective and Ding injective modules over trivial ring extensions
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 903-919
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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.
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.
DOI : 10.21136/CMJ.2023.0351-22
Classification : 16D40, 16D50, 16E05
Keywords: trivial extension; Ding projective module; Ding injective module 
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Mao, Lixin. Ding projective and Ding injective modules over trivial ring extensions. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 903-919. doi: 10.21136/CMJ.2023.0351-22

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