Analytic extensions of a commutative ring
Let R be a commutative ring. Then \(I_ c(R):=\{a\in R:\) there exists a retraction \(Rt\to R\) with \(t\mapsto a\}\) is an ideal and consists, roughly speaking, of those elements of R which may be inserted in formal power series over R. This was studied earlier by the first two authors [see e.g. J. Algebra 67, 110-128 (1980; Zbl 0472.13009)]. If \(R\to S\) is a homomorphism of commutative rings one is interested in the inclusions between \(I_ c(R)\) and the counterimage of \(I_ c(S).\) An ideal \(\Omega\) \(\subset R\) is called closed if \((I_ c(R)+\Omega)/\Omega \subset I_ c(R/\Omega)\). For the closedness some sufficient conditions are given as well as some properties justifying this notation. All finitely generated ideals are closed, and R is Noetherian if and only if all ideals in R are closed. An ideal \(j_ R(\Omega)\) is defined for any ideal \(\Omega\) \(\subset R\) as the union of all \(R\cap \sum^{n}_{i=1}(X_ i-\omega_ I)\quad RX_ 1,...,X_ N \) with \(n\in {\mathbb{N}}\) and \(\omega_ i\in \Omega\). Its importance is in that \(\Omega \subset I_ c(R)\) if and only if \(j_ R(\Omega)=0\). In any case \((\cap \Omega^ n)^ 2\subset j_ R(\Omega)\subset \cap \Omega^ n.\) Furtheron commutative ring extensions \(S\supset R\) are considered where S is supposed to be finitely generated as an R-module. For \(I_ c(R)\subset R\cap I_ c(S)\) and for the reverse inclusion several sufficient conditions are shown. Finally Noetherian rings are studied; here the authors prove the formula \(I_ c(S)=\sqrt{I_ c(R)\cdot S}\) and a Weierstrass preparation theorem for \(RX_ 1,...,X_ n\).
scientific article; zbMATH DE number 4099452