Flat modules and rings finitely generated as modules over their center

Abstract

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions: (1) A is a right or left distributive semiprime ring; (2) for any maximal idealM of a subringR central inA, the ring of quotientsA M is a finite direct product of semihereditary Bezout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields; (3) all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.