We extend the ring-theoretic concept of going down to algebraic frames and coherent maps. We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1. An application to rings yields a characterization of von Neumann regular rings that appears to have hitherto been overlooked. Namely, a commutative ring $A$ with identity is von Neumann regular if and only if $Ann(I)+P=A$, for every prime ideal $P$ of $A$ and any finitely generated ideal $I$ of $A$ contained in $P$.
Primary Language | English |
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Subjects | Mathematical Sciences |
Journal Section | Mathematics |
Authors | |
Publication Date | December 8, 2019 |
Published in Issue | Year 2019 Volume: 48 Issue: 6 |