Generalized divisor topology of commutative rings

Abstract

Let $R$ be a commutative ring with nonzero identity and let $R^{\#}$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^{\#})$. Its basic open sets are \[ B_a=\{[b]\in EC(R^\#): b\mid a^n \text{ for some }n\geq 1\}. \] The relation \[ [b]\in B_a \quad\Longleftrightarrow\quad \sqrt{aR}\subseteq\sqrt{bR} \] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.