A note on Itoh (e)-Valuation Rings of and Ideal

Let $I$ be a regular proper ideal in a Noetherian ring $R$, let $e \ge 2$ be an integer, let $\mathbf T_e = R[u,tI,u^{\frac{1}{e}}]' \cap R[u^{\frac{1}{e}},t^{\frac{1}{e}}]$ (where $t$ is an indeterminate and $u =\frac{1}{t}$), and let $\mathbf r_e = u^{\frac{1}{e}} \mathbf T_e$. Then the Itoh (e)-valuation rings of $I$ are the rings $(\mathbf T_e/z)_{(p/z)}$, where $p$ varies over the (height one) associated prime ideals of $\mathbf r_e$ and $z$ is the (unique) minimal prime ideal in $\mathbf T_e$ that is contained in $p$. We show, among other things: (1) $\mathbf r_e$ is a radical ideal if and only if $e$ is a common multiple of the Rees integers of $I$. (2) For each integer $k \ge 2$, there is a one-to-one correspondence between the Itoh (k)-valuation rings $(V^*,N^*)$ of $I$ and the Rees valuation rings $(W,Q)$ of $uR[u,tI]$; namely, if $F(u)$ is the quotient field of $W$, then $V^*$ is the integral closure of $W$ in $F(u^{\frac{1}{k}})$. (3) For each integer $k \ge 2$, if $(V^*,N^*)$ and $(W,Q)$ are corresponding valuation rings, as in (2), then $V^*$ is a finite integral extension domain of $W$, and $W$ and $V^*$ satisfy the Fundamental Equality with no splitting. Also, if $uW = Q^e$, and if the greatest common divisor of $e$ and $k$ is $d$, and $c$ is the integer such that $cd = k$, then $QV^* = {N^*}^c$ and $[(V^*/N^*):(W/Q)] = d$. Further, if $uW = Q^e$ and $k = qe$ is a multiple of $e$, then there exists a unit $\theta_{e} \in V^*$ such that $V^* = W[\theta_{e},u^{\frac{1}{k}}]$ is a finite free integral extension domain of $W, QV^* = {N^*}^q, N^* = u^{\frac{1}{k}}V^*$, and $[V^*:W] = k$. (4) If the Rees integers of $I$ are all equal to $e$, then $V^* = W[\theta_e]$ is a simple free integral extension domain of $W, QV^* = N^* = u^{\frac{1}{e}}V^*$, and $[V^*:W] = e = [(V^*/N^*):(W/Q)]$.