"Almost" ring homomorphism
Solution 1:
Fact. If a group $G$ is a union of two subgroups $G = G_1 \cup G_2$, then $G=G_1$ or $G=G_2$.
We use this two times. Let $a \in R$. Then $W_a$ and $V_a$ are additive subgroups of $R$ with union $R$. Hence, $W_a=R$ or $V_a=R$. Now, $\{a \in R : W_a=R\}$ and $\{a \in R : V_a=R\}$ are additive subgroups of $R$ with union $R$. Hence, $\{a \in R : W_a=R\}$, i.e. $\phi$ is a homomorphism, or $\{a \in R :V_a=R\}$, i.e. $\phi$ is an anti-homomorphism.