Closures and co-closures attached to FCP ring extensions

Abstract

The paper deals with ring extensions $R\subseteq S$ and the poset $[R,S]$ of their subextensions, with a special look at FCP extensions (extensions such that $[R,S]$ is Artinian and Noetherian). When the extension has FCP, we show that there exists a co-integral closure, that is a least element $\underline R$ in $[R,S]$ such that $\underline R \subseteq S$ is integral. Replacing the integral property by the integrally closed property, we are able to prove a similar result for an FCP extension. The radicial closure of $R$ in $S$ is well known. We are able to exhibit a suitable separable closure of $R$ in $S$ in case the extension has FCP, and then results are similar to those of field theory. The FCP property being always guaranteed, we discuss when an extension has a co-subintegral or a co-infra-integral closure. Our theory is made easier by using anodal extensions. These (co)-closures exist for example when the extension is catenarian, an interesting special case for the study of distributive extensions to appear in a forthcoming paper.