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!