Reduction of Ideals Relative to an Artinian Module and the Dual of Burch’s Inequality

Let (A, 𝔪) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with ℓ(0 :M I) < ∞, then we show that for a minimal reduction J of I, (0 :M JI) = (0 :M I2) if and only if [Formula: see text] for all n ≥ 0. Moreover, we study the dual of Burch’s inequality. In particular, the Burch’s inequality becomes an equality if G(I, M) is co-Cohen-Macaulay.