2016/10/07 by Ryan C. Schwiebert · 1 citation
Computer Science · Decision Sciences · Mathematics · #Advanced Algebra and Logic #Closure (psychology) #Commutative property #Commutative ring #Complement (music) #Free monoid #Fuzzy and Soft Set Theory #Monoid #Ring (chemistry) #Rings, Modules, and Algebras #Semiprime ring #Syntactic monoid #math.RA #msc:06F05 #msc:16D99 #msc:16N60
paper · pdf · doi:10.1080/00927872.2016.1222401
published as Ryan C. Schwiebert (2016) The radical-annihilator monoid of a ring, Communications in Algebra, 45:4, 1601-1617 · 5 figures
openalex publication_date 2016/10/07 · openalex created_date 2016/10/14 · arxiv created 2018/03/01 · arxiv updated 2018/03/02 · openalex updated_date 2026/08/05
Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.