Let $A$ be a commutative, unital ring and let $M_i$ and $N_i$ be $A$-modules ($i=1,2)$.
If $\alpha_1:M_1\to N_1$ and $\alpha_2:M_2\to N_2$ are $A$-linear, then we get a unique $A$-linear map $$\alpha_1\otimes\alpha_2 : M_1\otimes_A M_2\to N_1\otimes N_2$$ which sends $$m_1\otimes m_2 \mapsto \alpha_1(m_1)\otimes\alpha_2(m_2)$$ for all $m_i \in M_i.$
Question: How much can we say about the kernel of this map?
Thoughts so far: It clear that if $m_1 \in \ker(\alpha_1),$ then $m_1\otimes y \in \ker(\alpha_1\otimes\alpha_2)$ for all $y \in M_2.$
Similarly, if $m_2 \in \ker(\alpha_2),$ then $x\otimes m_2 \in \ker(\alpha_1\otimes \alpha_2)$ for all $x \in M_1.$
This would lead me to guess that (perhaps under some extra conditions) we might have:
$$\ker(\alpha_1\otimes \alpha_2)\cong(\ker(\alpha_1)\otimes_A M_2)\oplus(M_1\otimes_A\ker(\alpha_2)).$$
Gut feeling: Since tensoring doesn't generally preserve injectivity, things might not be so clean...
I would be grateful if someone could tell me what results there are in this direction if any!
Many thanks :)