A simple proof of Hilbert basis theorem for $*_w$-Noetherian domains

Abstract

Let $D$ be an integral domain with quotient field $K$, $*$ a star-operation on $D$, GV$^*(D)$ the set of nonzero finitely generated ideals $J$ of $D$ such that $J_*=D$, and $*_w$ a star-operation on $D$ defined by $I_{*_w}=\{x \in K \mid Jx \subseteq I$ for some $J \in {\rm GV}^*(D)\}$ for all nonzero fractional ideals $I$ of $D$. In this article, we give a simple proof of Hilbert basis theorem for $*_w$-Noetherian domains.