Flat modules and rings finitely generated as modules over their center

A. A. Tuganbaev

Matematičeskie zametki, Tome 60 (1996) no. 2, pp. 254-277

Résumé

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 ring $A$ be a finitely generated module over its unitary central subring $R$. We prove the equivalence of the following conditions: (1) $A$ is a right or left distributive semiprime ring; (2) for any maximal ideal $M$ of a subring $R$ central in $A$, the ring of quotients $A_M$ is a finite direct product of semihereditary Bézout 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 ring $A$ are flat (right and left) modules over the ring $A$, and $A$ is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.