Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring and let $M$ be a $GV$-torsionfree $R$-module. Then $M$ is said to be a $w$-module if ${\rm Ext}_R^1(R/J, M)=0$ for any $J\in GV(R)$, and the $w$-envelope of $M$ is defined by $M_w=\{x\in E(M) | Jx\subseteq M \mbox{ for some } J\in GV(R)\}$. In this paper, $w$-modules over commutative rings are considered, and the theory of $w$-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of $w$-Noetherian rings and Krull rings.

Keywords: $GV$-ideal, $GV$-torsionfree module, $w$-module, $w$-Noetherian ring, Krull ring

MSC numbers: Primary 13A15, 13D99; Secondary 13E99, 13F05