I am trying to find out the $\ker(f\otimes g)$ where $f:M \rightarrow P$ and $g:N \rightarrow Q$ are $A$ linear maps where $A$ is not a field. So $(f\otimes g):M\otimes N\rightarrow P\otimes Q $ is $A$ linear map.The question is: can the following inclusion be proper?
$(\ker f\otimes N)\cup(M\otimes \ker g)\subset \ker(f\otimes g) $.
Is there any condition on the modules like flatness (other than vector space case) which forces the reverse containment ?
I am not getting anything,Help me.Thanks in advance.
$\ker(f) \otimes N$ doesn't have to be a submodule of $\ker(f \otimes g)$. But there is a canonical homomorphism $\ker(f) \otimes N \to \ker(f \otimes g)$. Similarly, we get a canonical homomorphism on $M \otimes \ker(g)$. Hence, we get a canonical homomorphism $\ker(f) \otimes N \oplus M \otimes \ker(g) \to \ker(f \otimes g)$. This turns out to be surjective when $f,g$ are surjective, because of the right exactness of the tensor product.
For more on this, see What is the kernel of the tensor product of two maps?