Induced map on cohomology for a product space

Question

Suppose we have connected CW complexes $X$, $Y$, and $Z$, and a map $f:X \times Y \to Z$, and that $f$ maps $X \times \{y_0\}$ to $\{z_0\}$. Suppose furthermore that we take coefficients from a free R-module $R$ and $H^k(Y;R)$ is a finitely-generated free R-module for all $k$. Then, given any $n$, $f$ induces a map $f^*:H^n(Z,R) \to H^n(X \times Y,R)$. $H^n(X \times Y,R)$ then splits as a sum $\sum_{i=0}^n H^i(X;R)\otimes_R H^{n-i}(Y;R)$ for R-module $R$. Given any $\alpha \in H^n(Z;R)$, is there a way to show that the $i=0$ summand in the splitting of $f^*(\alpha)$ is zero? I would greatly appreciate any help. Thank you.