First steps going down on algebraic frames

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Hacettepe Journal of Mathematics and Statistics
Volume:48 Issue:6
First steps going down on algebraic frames

First steps going down on algebraic frames

Pages : 1792-1807

View : 21 | Download : 9

Publication Date : 2019-12-08

Article Type : Research Paper

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 $Anninsert ignore into journalissuearticles values(I);+P=A$, 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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