On the integral degree of integral ring extensions

Abstract

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\mathrm{d}}_A(B)$, is defined as the supremum of the degrees of minimal integral equations of elements of $B$ over $A$. It is an invariant that lies in between ${\mathrm{d}}_K(L)$ and ${\mu }_A(B)$, the minimal number of generators of the $A$-module $B$. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if $A\subset B$ is simple; if $A\subset B$ is projective and finite and $K\subset L$ is a simple algebraic field extension; or if $A$ is integrally closed. Furthermore, $\mathrm{d}$ is semicontinuous if $A$ is noetherian of dimension $1$ and with finite integral closure. In general, however, $\mathrm{d}$ is neither sub-multiplicative nor upper-semicontinuous.