On a basic tensor product question I am trying to show that
$$
\mathbb{Z} / (10) \otimes \mathbb{Z} / (12) \cong \mathbb{Z}/(2)
$$
by defining a map
$$
h([a]_{10} \otimes [b]_{12}) = [ab]_2
$$
and extend it linearly. I am having trouble trying to prove that this map is 
well defined on $\mathbb{Z} / (10) \otimes \mathbb{Z} / (12)$. (I am using the 
constructive definition of tensor products for exercise). I would greatly appreciate any help!
Thanks!
 A: try to show in general that: let   $A$ a commutative ring and $I$ , $J$ two ideals, then $A/I\otimes A/J\simeq A/(I+J)$. And remark that $12\mathbb{Z}+10\mathbb{Z}=2\mathbb{Z}$.   
A: Let's recall the constructive definition of the tensor product: Suppose $M$ and $N$ are $A$-modules. Let $F$ be the free $\mathbb{Z}$-module on the set of symbols $\{e_{m,n} \ | \ (m,n)\in M\times N\}.$ Let $I$ be generated by $e_{m+m',n} - e_{m,n} - e_{m',n}$ etc (I'm sure you know the other generators). Then $F/I$ is a tensor product. 
You want to define an $A$-linear map $h: F/I \to \mathbb{Z}/(2).$ To do this, first define a map $h^*: F\to \mathbb{Z}/(2).$ Since $F$ is a free module, you get an $A$-module homomorphism by assigning where the basis elements $e_{m,n}$ go, then extending linearly. Let's define $h^*( e_{m,n} ) = mn.$ 
Now check that $h^*$ vanishes on $I.$ For example: $$h^*( e_{m+m',n} - e_{m,n} - e_{m',n}) = (m+m')n - mn - m'n =0.$$ A similar computation for the other generators of $I$ shows $I\subseteq \ker h^*.$ 
This means that $h^*:F\to \mathbb{Z}/(2)$ descends to a well defined $A$-linear map $h:F/I \to \mathbb{Z}/(2)$ defined by $h(x+I) = h^*(x).$ This is a special case of a  general fact that you should prove if you haven't seen before: If $\varphi: M\to N$ is an $A$-module homomorphism and $L\subseteq M$ is a submodule inside the kernel of $\varphi$ then $\phi: M/L \to N$ defined by $\phi(m+L) = \varphi(m)$ is a well defined $A$-module homormorphism. 
This shows that the map you defined is a well defined $A$-module homomorphism, which is the part you said you needed help with. If you need any help showing the map is an isomorphism feel free to leave a comment saying so. 
