Localization with Respect to Endomorphisms

Abstract. We introduce the class of localizable monoids. It contains inverse monoids. Then we define localizations of monoids with respect to localizable submonoids of their monoid of endomorphisms. These constructions can be applied to a category of left modules or to a category of A -rings. As a result, we are able to invert endomorphisms within the original category, unlike inversive localizations of Cohn's type which need a base change.