Basic question about tensor products I almost feel embarrassed to ask this, but I am trying to learn about tensor products (for now over Abelian groups). Here is the definition given:
Let $A$ and $B$ be abelian groups. Their tensor product, denoted by $A \otimes B$, is the abelian group having the following presentation
Generators: $A \times B$ that is, all ordered pairs $(a,b)$ 
Relations: $(a+a',b)=(a,b)+(a',b)$ and $(a,b+b')=(a,b)+(a,b')$ for all $a,a' \in A$ and $b,b' \in B$
So from this, why is $a \otimes 0 = 0$? Looks to me like if $b$ is zero, then any $a,a' \in A$ will still satisfy the relations. I'm just after a simple explanation, then hopefully once that makes sense, it will all make sense!

You are missing the relations $(na,b)=(a,nb)=n(a,b)$ for all $n\in\mathbb{Z}$ (recall that abelian groups are $\mathbb{Z}$-modules). Thus 
$$a\otimes 0_B=a\otimes (0_{\mathbb{Z}}\cdot 0_B)=(0_{\mathbb{Z}}\cdot a)\otimes 0_B=0_A\otimes 0_B=0.$$
