regular ring is flat. The converse is also true. Suppose B is flat. Then the sequence is pure exact if and only if C is flat. From this one can deduce that pure submodules of flat modules are flat. Suppose C is flat. Then B is flat if and only if A is flat. If is pure-exact, and F is a finitely presented R-module, then every homomorphism from F to C can be lifted to B, i.e. to every u : F → C there exists v : F → B such that gv=u. References Module theory