Maximal Quotient Rings1

Let R be an associative ring in which an identity element is not assumed. A right quotient ring of P is an overring 5 such that for each aQS there corresponds rQR such that arQR and ar 9*0. A theorem of R. E. Johnson [l ] states that R possesses a right quotient ring S which is a (von Neumann) regular ring if and only if P has vanishing right singular ideal. In this case P possesses a unique (up to isomorphism over P) maximal right quotient ring S, and 5 is regular and right self-injective (Johnson-Wong [l]). It is easy to see that 5 is the injective hull of P, considering both rings as right P-modules in the natural way. Thus, each right ideal I oí R has an injective hull Îr contained in 5. In this notation, S=R¡¡, and we use R to denote the maximal right quotient ring of P hereafter. By the results of Johnson [2], Îr can be characterized in two ways: (a) Ir is the unique maximal essential extension of I contained in the right P-module P. (b) Îr is the principal right ideal of R generated by I. Since Îr is therefore a right ideal of R, A = Homg (ÎR, ÎR) is defined. Setting r = HomÄ (7, I), one of our main results (Theorem 2) states that f = Homg (Îr, Îr) =A. This means that T has vanishing right singular ideal, and that A is the maximal right quotient ring of r. Since Îr is a principal right ideal in the regular ring R, there exists an idempotent eQR such that îR — eÈ.. Then, of course, AÇ^eRe, and it is natural to investigate the relationship between eRe and K = eRei\R. In general, it is too much to hope that eRe = K, since it is possible that K = Of or some nonzero eQR. Nevertheless, under the assumption that R is also a left quotient ring of P, or in case e is a primitive idempotent satisfying eReC\R9*0, we establish (Theorem 3) that K has vanishing right singular ideal, and that eRe = K ( = the maximal right quotient ring of A.) In any case, for any nonzero idempotent eGPi eRe is the maximal right quotient ring of eRe. Since we are not restricting ourselves to rings with identity, we say