On the product of a module by an ideal (Q909721)

If R is a commutative ring with unit and if I and J are multiplication ideals in R (in the sense of D. D. Anderson), then so is IJ. Moreover, if I and J are both projective, respectively flat, then IJ is projective, respectively flat, as well. The author generalizes this result to R-modules: if M is a finitely generated multiplication module and I a projective, respectively flat, ideal in R such that \(I\cap Ann(M)=0\), then MI is projective, respectively flat. Note that if the last condition does not hold, then the result is not necessarily valid anymore.