$\newcommand{\pdim}{\operatorname{pdim}}$If $\pdim_A M$ is the projective dimension of $M$ as an $A$-module how can i prove that if $A/I=A'$ then $$\pdim_A M\leq \pdim_A A' + \pdim_{A'} M$$ If the left summand are finite and$M$ is an $A'$-module?
On a side note: if $M$ can be an $A$ module with two different operation, are there properties or something else of $M$ that doesn't depend on the operation i chose?