Ding projective and Ding injective modules over trivial ring extensions

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⊗RM⊗RX→M⊗αM⊗RX→αX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M{\otimes _R}M{\otimes _R}X\mathop \to \limits^{M \otimes \alpha} M{\otimes _R}X\mathop \to \limits^\alpha X$$\end{document} 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.