Flat and Multiplication Modules

Abstract

All rings are assumed to be associative and with nonzero identity element, and all modules are assumed to be unitary. A ring is said to be right (resp., left) invariant if all right (resp., left) ideals of it are ideals. Expressions such as “an invariant ring” mean that the corresponding right and left conditions hold. A right module M over a ring A is called a multiplication module if for every submodule N of M , there exists an ideal B of the ring A such that N = MB. A module MA is said to be flat if for every left A-module L, the natural group homomorphism M ⊗A L is a monomorphism. If all submodules of any right or left module over a ring A are flat, then one says that the weak global dimension of the ring A does not exceed 1; in this case, we write w.gl.dim(A) ≤ 1 for brevity. In [7], it is proved that every faithful multiplication module over a commutative ring is a flat module. If A is a commutative ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive semiprime ring [5]. If A is an invariant semiprime ring, then w.gl.dim(A) ≤ 1 if and only if A is a distributive ring [13, Theorem 7.31]. The main result of the present paper is Theorem 1.