Infinite group-graded rings, rings of endomorphisms, and localization

Abstract

The paper is divided into four sections. The first section gives a matricial description R ▿ G of the ring End R- gr ( U), which contains as a unital subring the smash product R ̃ #G, R= ⊕ x ∈ GR x being a group-graded ring and =⊕ x ∈ G R( x) the canonical generator of the category R- gr. Section 2 proves mainly the following two facts: for each subring B of R ▿ G containing R̃#G, R- gr is equivalent to a quotient category of B- mod, and the ring R ▿ G is isomorphic to a ring of quotients of B. Section 3 investigates the structure of graded endomorphism rings of the type END R (⊕ x ∈ G M( x)), where M∈ R- gr is a finitely generated R-module. In the last section it is shown that R- mod is equivalent to a quotient category of the category END R ( U)- mod, and End R ( U) is isomorphic to a ring of quotients of the ring END R ( U).