COMMUNICATIONS IN ALGEBRA, 2(5), 403–427 (1974) PERFECT QUOTIENT FUNCTORS John A. Beachy Department of Mathematical Sciences Northern Illinois University DeKalb, Illinois 60115 Recently there has been considerable interest in extending to non- commutative rings the techniques of localization which have proved to be useful in the study of commutative rings. The notion of a quo- tient category, as utilized by Gabriel [5], has played a fundamental role in one approach to the problem. The constructions considered by Gabriel include quotient categories determined either by localizing Serre subcategories, certain filters of left ideals, or injective modules. (A special case of the latter method had been introduced by Findlay and Lambek [4].) Further equivalent notions are the torsion radicals of Maranda [12] (called idempotent kernel functors by Goldman [6]) and the (hereditary) torsion theories of Dickson [3] and Lambek [8]. The reader is also referred to the Walkers’ paper [20] and the recent expositions of Lambek [9,11], Morita [14], and Stenstr¨ om [18], which contain extensive bibliographies. A major difficulty in this approach is that the quotient category in general need not coincide with the category of modules over the cor- responding ring of quotients. This paper gives an expository account 403