On rings of which any one-sided quotient rings are two-sided

Abstract

0. Introduction and preliminaries. Following R. E. Johnson [1] we assume in this paper that any rings we shall be concerned with satisfy either one or both of the following conditions: (Jf) If the right annihilator of a left ideal A is nonzero, then there exists a nonzero left ideal B such that A nB =0. (J) =the right left symmetry of (J1).I We say that a ring is a Ji-ring, a Jf-ring or a J-ring if it satisfies (Ja), (J,) or both of them. A module A is called an essential extension of a submodule B if BnCFCO for every nonzero submodule C of A. A module is said to be injective if it is a direct summand of every extension module. It is well known that every module M has a maximal essential extension ff. A is injective, and is unique to within an isomorphism over M. Let S be a Ji-ring. Then we can define the multiplication in the maximal essential extension 3 of the left S-module S such a way that (i) 3 forms a ring and (ii) the multiplication coincides, on SX3, with the scalar multiplication. This ring is unique up to an isomorphism over S, and is denoted by Sz. As is known, SI is regular (in the sense of von Neumann); and is left self injective, that is, injective as a left module over itself. An extension ring T of a Ji-ring S is called a left quotient ring of S if the left S-module T is an essential extension of the left S-module S. It is also known that every left quotient ring of S is isomorphic, over S, to a subring of 3;. Thus, Sl is the maximal left quotient ring of S. We define similarly a right quotient ring and the maximal right quotient ring ST of a Jr-ring S. For any J-ring S it is easily seen that the following conditions are equivalent: (i) There exists an extension ring T of S with the properties that (a) it is regular (both left and right) self injective, and (b) every nonzero one-sided S-submodule of T has a nonzero intersection with S. (ii) Every left quotient ring of S is a right quotient ring of S, and every right quotient ring of S is a left quotient ring of S. In this case any maximal left quotient ring of S and any maximal