Ding modules and dimensions over formal triangular matrix rings (Q2697591)

Summary: Let \(T = \begin{pmatrix} A & 0 \\ U & B \end{pmatrix}\) be a formal triangular matrix ring, where \(A\) and \(B\) are rings and \(U\) is a \((B, A)\)-bimodule. We prove: (1) If \(U_A\) and \({}_B U\) have finite flat dimensions, then a left \(T\)-module \(\binom{M_1}{M_2}_{\varphi^M}\) is Ding projective if and only if \(M_1\) and \(M_2 / \mathrm{im} (\varphi^M)\) are Ding projective and the morphism \(\varphi^M\) is a monomorphism. (2) If \(T\) is a right coherent ring, \({}_B U\) has finite flat dimension, \(U_A\) is finitely presented and has finite projective or \(\mathrm{FP}\)-injective dimension, then a right \(T\)-module \((W_1, W_2)_{\varphi_W}\) is Ding injective if and only if \(W_1\) and \(\ker (\widetilde{\varphi_W})\) are Ding injective and the morphism \(\widetilde{\varphi_W}\) is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a \(T\)-module.