Algebra universalis

, Volume 74, Issue 3–4, pp 411–424 | Cite as

Covering maximal ideals with minimal primes

  • Themba Dube
  • Oghenetega Ighedo


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.

Keywords and phrases

frame ring of continuous real-valued functions on a frame f-ring maximal ideal minimal prime ideal UMP-ring 

2010 Mathematics Subject Classification

Primary: 06D22 Secondary: 54E17 13A15 18A40 


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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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