Abstract
Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M such that MR, RM, (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 \(M{\otimes _R}M{\otimes _R}X\mathop \to \limits^{M \otimes \alpha} M{\otimes _R}X\mathop \to \limits^\alpha X\) 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.
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The author wants to express his gratitude to the referee for his/here very helpful comments.
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This research was supported by NSFC (12171230, 12271249) and NSF of Jiangsu Province of China (BK20211358).
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Mao, L. Ding projective and Ding injective modules over trivial ring extensions. Czech Math J 73, 903–919 (2023). https://doi.org/10.21136/CMJ.2023.0351-22
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DOI: https://doi.org/10.21136/CMJ.2023.0351-22