{"title":"Building Noetherian and Non‐Noetherian Integral Domains Using Power Series","authors":"W. Heinzer, C. Rotthaus, S. Wiegand, Jim Huckaba","doi":"10.1201/9780429187902-19","DOIUrl":"https://doi.org/10.1201/9780429187902-19","url":null,"abstract":"In this mainly expository article we describe a technique, dating back at least to the 1930s, which uses power series, homomorphic images and intersections involving a Noetherian integral domain R and a homomorphic image S of a power series ring extension of R to obtain a new integral domain A. Here A has the form A := L∩ S, where L is a field between the fraction field of R and the total quotient ring of S. We give in certain circumstances necessary and sufficient conditions for A to be computable as a nested union of subrings of a specific form. We also prove that the Noetherian property for the associated nested union is equivalent to a flatness condition. We present several examples where this flatness condition holds, and other examples where it fails to hold. In the first case this produces a Noetherian integral domain and in the second case a non-Noetherian domain.","PeriodicalId":411392,"journal":{"name":"Ideal Theoretic Methods in Commutative Algebra","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133242727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Complete Integral Closure of the Rees Algebra","authors":"Stefania Gabelli, Anna Guerrieri","doi":"10.1201/9780429187902-16","DOIUrl":"https://doi.org/10.1201/9780429187902-16","url":null,"abstract":"","PeriodicalId":411392,"journal":{"name":"Ideal Theoretic Methods in Commutative Algebra","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127937095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}