CLOSURES AND CO-CLOSURES ATTACHED TO FCP RING EXTENSIONS.

The paper deals with ring extensions R ⊆ 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 R in [R, S] such that R ⊆ S is integral. 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. [ABSTRACT FROM AUTHOR]