A note on the dual of Burch's inequality (Q1567654)

The author improves the main result of \textit{I. Nishitani} [J. Pure Appl. Algebra 96, 147-156 (1994; Zbl 0812.13008)] and shows that if \(A\) is a non-zero Artinian module over a commutative ring \(R\) and \({\mathfrak a} \subseteq {\mathfrak b}\) are ideals of \(R\) such that the annihilator \((0:{_A {\mathfrak b})}\) is not zero then the dual of Burch's inequality \[ S_{\mathfrak b}( {\mathfrak a},A)\leq \text{Kdim}_R(A)- \text{width}_{\mathfrak b} (0:{_A{\mathfrak a}^i)}\quad (i\geq 0) \] holds, where \(S_{\mathfrak b}({\mathfrak a},A)\) is the dual analytic spread of \({\mathfrak a}\) at \({\mathfrak b}\) relative to \(A\) and \(\text{Kdim}_R(A)\) is the dual Krull dimension of \(A\).