Flat regular quotient rings

Abstract

0. Introduction and notation. In this paper we study the condition that the maximal right quotient (MRQ) ring Q [10, p. 106] of a right nonsingular ring R with 1 is flat as a left R-module. It is known [11, p. 134] that if Q is the classical right quotient ring of R, then Q is flat as a left R-module. This is not always the case with the MRQ ring of R: in ?2 we obtain an ideal theoretic characterization (Theorem 2.1) and a module theoretic characterization (Theorem 2.2) of a right nonsingular ring R, all of whose regular right quotient rings are flat as left R-modules; we also indicate the existence of a class of commutative rings R, whose singular ideal is zero and for which the maximal quotient ring is not R-flat. Throughout this paper R denotes an associative ring with identity. A right R-module M is denoted MR; all R-modules are unitary. Let NR and MR be modules such that NR. (MR. We say that NR is large in MR (MR is an essential extension of NR) if NR intersects nontrivially every nonzero submodule of MR. A right ideal I or R is large in R if IR is large in RR. For any module MR, L(MR) denotes the lattice of large submodules of MR. Let MR be a module. We denote by Z(MR) the singular submodule of MR. If for any x E M we set (0: x) ={r E R I xr =0}, then