First steps going down on algebraic frames

Authors : Themba DUBE

Pages : 1792-1807

Doi:10.15672/HJMS.2018.638

View : 13 | Download : 5

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