Topics on strict closure of rings

TL;DR

This paper explores advanced topics in the theory of strict closure of rings, including construction methods, inheritance properties under flat homomorphisms, and the relationship with Arf closure, supported by illustrative examples.

Contribution

It advances the theory of strict closure by addressing construction, inheritance, and comparison with Arf closure, extending Lipman's foundational work.

Findings

Constructed new methods for strict closure

Established conditions for inheritance under flat homomorphisms

Identified when Arf closure coincides with strict closure

Abstract

In 1971, J. Lipman introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns construction of the closure, and the second one is the study regarding the question of whether the strict closedness is inherited under flat homomorphisms. We finally handle the question of when the Arf closure coincides with the strict closure. Examples are explored to illustrate our theorems.