In a local Cohen-Macaulay ring (A, m), we study the Hilbert function of an m-primary ideal I whose reduction number is two. It continuous work papers Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With some conditions, show inequality e1(I) ? e0(I) ? ?A(A/I) + e2(I) coefficients, which converse Sally Itoh. We also relations between coefficients depth associated graded ring.

The homological property of the associated graded ring an ideal is important problem in commutative algebra and algebraic geometry. In this paper we explore almost Cohen–Macaulayness stretched $\mathfrak{m}$-primary ideals case where reduction number attains minimal value a Cohen–Macaulay local $(A,\mathfrak{m})$. As application, present complete descriptions with small number.

In this article, we introduce an invariant of Cohen-Macaulay local rings in terms the reduction number canonical ideals. The can be defined arbitrary and it measures how close to being Gorenstein. First, clarify relation between almost Gorenstein nearly by using dimension one. We next characterize idealization trace ideals over invariant. It provides better prospects for a result on property id...