Example of a bilinear form of abelian groups

Question

Let $X$ and $Y$ be abelian groups. Then, a $Y$-valued bilinear form on $X$ is a $\mathbb{Z}$-module homomorphism $$\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$$ How does this relate to the standard notion of a bilinear map $f: X \times X \rightarrow Y$ where $f(x_1 + x_2, x_3) = f(x_1,x_3) + f(x_2,x_3)$ and so on?

Where does the tensor product come in to this?

Could anyone provide an example of a $Y$- valued bilinear form on $X$, where $Y$ and $X$ are abelian groups?

Answer 1

The function $X\times X\to Y$ prescribed by: $$(x_1,x_2)\mapsto\alpha(x_1\otimes x_2)$$ is bilinear.

Conversely if $f:X\times X\to Y$ is bilinear then a unique $\mathbb Z$-module homomorphism $\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$ exists such that: $$f(x_1,x_2)=\alpha(x_1\otimes x_2)$$

So there is a one-to-one correspondence between bilinear functions $f:X\times X\to Y$ and $\mathbb Z$-module homomorphisms $\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$.