Functorial Rings of Quotients I

Abstract

The term “ring of quotients” is meant in the classical sense: arising from the inversion of members of a subsemigroup of regular elements. We study functors which assign to each object in a category A of commutative rings with identity, a ring of quotients. (The total ring of quotients A R is not functorial.) There is a maximum such, A H , obtained by inverting all the elements which remain regular under any non-zero A-morphism. Under reasonable hypotheses, A H is a categorical reflection and the maximum subobject of A R over which all homomorphisms from A to B R ’s extend; and, the subcategory of all A R ’s is the epireflective hull of all B R ’s. For f-rings, A H is the closure under bounded inversion. For archimedean f-rings, it is the simultaneous closure under bounded inversion and countable l-inversion. For rings of real-valued functions, A H is the closure under inversion of functions without zeroes. ( A sequel will treat the analogue for the “complete” ring of (Johnson-Utumi) quotients.)