The shells of Vaggione: some comments and questions

Abstract

This article explores Vaggione's generalization of the author's notion of shell. Examples show that it is definitely more general. Categorical and polynomial equivalence reduce its essential generality to the possibility of adjoining new operations. These additional operations require additional clauses in the definitions of factor element and ideal, which correspond to the notion of factor congruence, all three of which form isomorphic Boolean algebras. As is customary one constructs a sheaf. Thus any shell is isomorphic to the shell of all global sections of the sheaf over the Boolean space of all prime ideals of factor objects. As a consequence, any Stone shell is isomorphic to the shell of all global sections where the stalks have no divisors of zero. Similarly, biregular shells yield simple stalks. These results encompass some classical theorems in ring theory.