Let and be rings with 1. Let be a right -module and let be an -bimodule. Prove that if is flat over and flat over , then is flat over .
Let be a short exact sequence of left -modules. Since is flat over , the natural sequence is short exact. Moreover, this is a sequence of left -modules since is an -bimodule. Since is flat over , is short exact. Using the natural isomorphisms, is short exact.
Thus is flat as a right -module.