Flat epimorphic extensions of rings

Abstract

An epimorphic extension of a ring R is a ring S for which there is a homomorphism 11: R ~ S which is both a monomorphism and an epimorphism in the category of rings. This paper is concerned with epimorphic extensions which are left-fiat in the sense that ~ induces on S the structure of a fiat left R-module; an example of such an extension is the (classical) ring of right quotients of R. Flat epimorphic extensions of commutative rings have been studied by Lazard [-8]; the more general situation in which ~ need not be monomorphic has been considered by Silver [-10]. Characterisations of left-fiat epimorphic extensions of a ring R are given in Section 3, from which it follows that such extensions are rings of right quotients of R in the sense of Utumi [121. Certain properties analogous to those possessed by classical rings of right quotients are also discussed. The principal result of the paper is Theorem 4.1, in which the existence of a left-fiat epimorphic extension P(R) of a ring R in which every other such extension can be uniquely embedded is established. It is shown, in Section 5, that, for a given ring R, P(R) is a semisimple artinian ring if and only if R contains no infinite direct sum of non-zero right ideals and the right singular ideal of R is zero. Finally, commutative semiprime rings are considered. It is shown that, if such a ring R contains all the idempotents of its complete ring Q (R) of quotients, then P(R) is the minimal regular subring of Q(R) which contains R. Examples of such rings are commutative semiprime rings which are integrally closed in Q (R) and commutative Baer rings. It has been drawn to my attention that some of the results of this paper are in the Notes [-3, 41 and [-91. Theorem 4.1 is announced, without proof, in [-91. A stronger result than Corollary 3.4 appears in [31 and [91 and Theorem 3.6 (iii) appears in [41. 2. Notation and Known Results