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?