Covering maximal ideals with minimal primes

Abstract

A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of β X is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which ${\mathcal{R} L}$ is a UMP-ring. All such frames are almost P-frames, and an Oz-frame is of this kind precisely when it is an almost P-frame. If L is perfectly normal (and hence if L is metrizable), then ${\mathcal{R} L}$ is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a ${\mathbb{Q}}$ -algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.