A ring-theoretic version of the Kuratowski 14-set problem

What is it about?

There is a classical problem in topology that asks this: given a topological space X, and a subset Y, how many distinct subsets could you possibly make from Y by using the set complementation and closure operations? In 1922, Kuratowski found the answer (14 subsets!). These two operations define mappings which generate a monoid, and the shape of that monoid can reflect some properties of the topological space. This paper explores the analogous problem for rings using the annihilator as a substitute for complementation, and radical as a substitute for closure. These also generate a monoid. Some nice results are derived about the monoid for nice classes of rings like Artinian rings and semiprime rings.

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Why is it important?

Between 1922 and 2015, work on similar problems largely involved generalizing the closure operator, or using more/different operations to generate the monoid. Very little, if anything, involved generalizing the complement operator or even left the venue of set operations. So in particular in this article, the novel ideas are to use the radical and annihilator operations operating on ideals (rather than all subsets), and to view the annihilator operator as a nontrivial generalization of the set-complementation operator. The article is able to comment on the interplay between the structure of the ring and its monoid just as the Kuratowski monoid of a topological space is used. The results are quite distinct from the classification of topological spaces via Kuratowski monoid.

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