Hello everyone, can someone tell me how to prove part(c).This is my atempt:
Define $\Phi: I\otimes I \longrightarrow \mathbb{Z}_2$ by $\Phi(p(x) \otimes q(x))= \frac{p(0)}{2}*q'(0)$, where $q'$ denotes the usual derivative. Now, i was able to show that:
$$\Phi[(p_1(x)+p_2(x)) \otimes (q_1(x)+q_2(x))] = (\frac{p_1(0)+p_2(0)}{2})*q_1'(0) + (\frac{p_1(0)+p_2(0)}{2})*q_2'(0) = \Phi[(p_1(x)+p_2(x)) \otimes q_1(x)]+ \Phi[(p_1(x)+p_2(x)) \otimes q_2(x)]$$
On the other hand, let $r(x)\in R = \mathbb{Z}[x]$. Can someone help me to prove that:
$$\Phi(r(x)p(x)\otimes q(x)) = r(x)\Phi(p(x)\otimes q(x))$$
Thanks