A simple proof of Hilbert basis theorem for ∗w-Noetherian domains

Abstract

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