Integral closure and generic elements

AbstractLet (R,m) be a formally equidimensional local ring with depthR⩾2 and I=(a1,…,an) an m-primary ideal in R. The main result of this paper shows that if I is integrally closed, then so is its image modulo a generic element, that is, if T=R[X1,…,Xn]/(a1X1+⋯+anXn), then IT¯=I¯T

Weakly Arf rings

In this paper, we introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.Comment: 57 page