The radical-annihilator monoid of a ring

ABSTRACT

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.

KEYWORDS:

• Annihilators
• Dual rings
• Kuratowski closure-complement problem
• Ordered monoids
• Radicals
• Semiprime rings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D99
• 16N60
• 06F05

Acknowledgments

The author wishes to thank the referee for his careful reading and numerous helpful recommendations. Thanks also to Greg Oman and Manfred von Willich for their time spent reading and giving feedback. Finally, thanks to Keith A. Kearnes for bringing Lemma 4.4 to light.