Journal of the
Korean Mathematical Society JKMS

Let $I$ be an ideal in a Gorenstein local ring $A$ with the maximal ideal $\frak{m}$. Then we say that $I$ is an $equimultiple$ $good$ $ideal$ in $A$, if $I$ contains a reduction $Q=(a_1, a_2, \cdots, a_s)$ generated by $s$ elements in $A$ and $\text{G}(I)=\oplus_{n\geq 0}I^n/I^{n+1}$ of $I$ is a Gorenstein ring with $\text{a}(\text{G}(I))=1-s$, where $s=\text{ht}_A I$ and $\text{a}(\text{G}(I))$ denotes the $\text{a}$-invariant of $\text{G}(I)$. Let $\mathcal {X}_A^s$ denote the set of equimultiple good ideals $I$ in $A$ with $\text{ht}_A I=s$, $\text{R}(I)=A[It]$ be the Rees algebra of $I$, and $\text{K}_{\text{R}(I)}$ denote the canonical module of $\text{R}(I)$. Let $a\in I$ such that $I^{n+1}=aI^n$ for some $n \geq 0$ and $\mu_A(I) \geq 2$, where $\mu_A(I)$ denotes the number of elements in a minimal system of generators of $I$. Assume that $A/I$ is a Cohen-Macaulay ring. We show that the following conditions are equivalent. \roster \item"{$(1)$}" $\text{K}_{\text{R}(I)} \cong \text{R}(I)_+$ as graded $\text{R}(I)$-modules. \item"{$(2)$}" $I^2=aI$ and $aA : I \in \mathcal{X}_A^1$.

Keywords: Rees algebra, associated graded ring, Cohen-Macaulay ring, Gorenstein ring, $\text{a}$-invariant

MSC numbers: Primary 13A30; Secondary 13H10