Let $R$ be a ring without identity.
Suppose that the multiplication $ \cdot : R \times R \rightarrow R $ is an abelian group homomorphism.
For $a, b \in R$ what can we conclude about the product of $a \cdot b$ ?
Let $m\colon R\times R\to R$ be the multiplication of $R$ and suppose $m$ is an abelian group homomorphism on the addition groups of $R\times R$ and $R$. Then, for any $a,b\in R$: $$ab = m(a,b) = m((a,0)+(0,b)) = m(a,0)+m(0,b) = 0+0 = 0.$$