If $R,R'$ are two rings, then $R\!\times\!0$ is a projective $R\!\times\!R'$-module, since it is a direct summand of a free module: $(R\!\times\!0)\oplus\!(0\!\times\!R')=R\!\times\!R'$.
What would be some sufficient conditions on $R$ and $R'$, so that $R\!\times\!0$ is not a free $R\!\times\!R'$-module? For example, if $R$ and $R'$ are unital, or commutative unital, does this suffice?