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.

中文翻译：

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正常对非交换环

本文将普通对的概念从交换环理论扩展到一对（缔合单元）环的上下文。这是通过使用Atterton引入的完整性概念来完成的。结果表明，如果\（R \ subseteq S \）是环，而\（D =（d_ {ij}）\）是一个\（n \ n n \）矩阵，并且在S中有项，则D是整数（在当且仅当每个\（d_ {ij} \）在R上是整数时，在\（n × n \）个矩阵的整个环上具有R的条目的Atterton感。如果\（R \ subseteq S \）是带有\ [n \ times n \]的相应完整环的环矩阵\（R_n \）和\（S_n \），则\（（R_n，S_n）\）是正常对，并且仅当（R，  S）是正常对时。给出了非交换（实际上是全矩阵）对的\（（\ Lambda，\ Gamma）\）对的例子，使得\（\ Lambda \ subset \ Gamma \）是（而不是）最小环延期; 可以进一步安排\（（\ Lambda，\ Gamma）\）是正常对或\（\ Lambda \ subset \ Gamma \）是整数扩展。