Normal pairs of noncommutative rings

Abstract

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $R \subseteq S$ are rings and $D=(d_{ij})$ is an $n\times n$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $n\times n$ matrices with entries in R if and only if each $d_{ij}$ is integral over R. If $R \subseteq S$ are rings with corresponding full rings of $n\times n$ matrices $R_n$ and $S_n$, then $(R_n,S_n)$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $(\Lambda , \Gamma )$ of noncommutative (in fact, full matrix) rings such that $\Lambda \subset \Gamma$ is (resp., is not) a minimal ring extension; it can be further arranged that $(\Lambda , \Gamma )$ is a normal pair or that $\Lambda \subset \Gamma$ is an integral extension.