Flat modules and rings finitely generated as modules over their center (Q1387318)

The distributivity of the ring \(A\) is investigated in the case when \(A\) satisfies the condition: (*) \(A\) is a finitely generated module over a central unitary subring \(R\). This article contains many facts on such rings, in particular the relations with rings of quotients and heredity of classes of flat modules are revealed. Among the main results the following two theorems are decisive. I. For a ring \(A\) with (*) the following conditions are equivalent: 1) \(A\) is a right distributive semiprime ring; 2) \(A\) is a left distributive semiprime ring; 3) for any ideal \(M\in\max(R)\) the ring of quotients \(A_M\) is the finite direct product of semihereditary uniform Bezout domains whose quotient rings by their Jacobson radicals are finite direct products of skew fields. II. If \(A\) is a left distributive ring with (*) then: 1) \(A\) is a distributive reduced ring over which all submodules of flat right modules and flat left modules are flat; 2) any prime quotient ring of the ring \(A\) is a semihereditary order in a skew field.