Rings of quotients and Morita contexts

Abstract

The Morita context which has been introduced in [9] was used since to prove Wedderburn theorem on the structure of simple rings [1] and in a disguised form it has been the basic tool in the study of primitive rings with a minimal left ideal (e.g. [6, p. 75]). Other applications, though not stated in an explicit form, can be found in various places. In the present paper we use the Morita context, to obtain various results: Goldie's theorem [2, 3] on the ring of quotients of semi-prime rings, following some ideas of Procesi (e.g. [5, p. 66]) and as a speciliazation one obtains Wedderburn's structure theorems of semi-simple artinian rings. The same methods are very useful in studying the ring of endomorphisms of modules V and in particular we obtain Zelmanowitz result [14] on the structure of Hom( V, V), if V is torsionless and finitely generated, and R is prime or semi-prime. Also, obtained are the facts that ring of endomorphisms of primitive rings are primitive [4, 10] and information on the radicals of ring of endomorphisms.