A note on integral closure

Abstract:

Let R be an integrally closed domain and ${x_i},{y_j}(1 \leqq i \leqq n,1 \leqq j \leqq m)$ R-sequences. Let \[ T = R[x_1^{{\alpha _1}} \cdots x_n^{{\alpha _n}}/y_1^{{\beta _1}} \cdots y_m^{{\beta _m}}],\] where the ${\alpha _i}$ and ${\beta _j}$ are positive integers. If T is integrally closed then \begin{equation}\tag {$*$}{\alpha _1} = \cdots = {\alpha _n} = 1\quad {\text {or}}\quad {\beta _1} = \cdots = {\beta _m} = 1.\end{equation} $( ^\ast )$ is sufficient for T to be integrally closed in the following cases: (1) R is Noetherian and the $({x_i},{y_j})R$ are distinct prime ideals, (2) R is a polynomial ring over an integrally closed domain and the ${x_i}$ and ${y_j}$ are indeterminates.