First steps going down on algebraic frames

Abstract

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 $P$.

Keywords

• Algebraic frame,Sublocale,Prime element,Going-down,Commutative ring,Dimension

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