Homological dimension of a class of special modules over a commutative ring (Q1814719)

\textit{P. Jothilingam} and \textit{S. Mangayarcarassy} have proved the following theorem [in Commun. Algebra 21, No. 2, 675-678 (1993; Zbl 0769.13004)]: If \(R\to A\) is a homomorphism of commutative rings with \(A\) Noetherian, self-injective ring, then \(A\) is \(R\)-flat if and only if \(A\) is \(R\)-injective. The author, in this note, generalizes the above theorem. He proves the following theorem: If \(R\to A\) is a homomorphism of commutative rings with \(A\) Noetherian and if \(N\) is a finitely generated injective \(A\)-module such that \(\max A\subseteq \text{Supp}(N)\), then the flat dimension of \(A\) as \(R\)-module coincides with the injective dimension of \(N\) as \(R\)-module. In proving this theorem the author proves a lemma in which a theorem of Bass-Roberts is used. But proper reference for this theorem is not given.