Extending a bilinear form to the direct sum

Question

Let $R$ be a ring with unity and let $V$ be a left $R$-module and let $A$ be an abelian group.

Then, an $A$-valued bilinear form on $V$ is $$\beta \in Bil(V,A) = {\{ \beta: V \times V \rightarrow A : \beta \hspace{2mm} \text{is} \hspace{2mm} \mathbb{Z}-\text{bilinear}}\}$$ Now, how do I extend this to a bilinear form on $V^n$, where $V^{n}$ is the direct sum of $n$ copies of $V$?

Answer 1

This is more of a comment, but I do not have sufficient reputation.

Why not do it coordinate-by coordinate? That is, define $ V^n \times V^n \rightarrow A $ by

$$ ((v_1,\dots,v_n),(w_1,\dots,w_n)) \mapsto \beta(v_1,w_1) + \beta(v_2,w_2) + \cdots + \beta(v_n,w_n) $$

You can check immediately that this defines an $A$-valued $\mathbb{Z}$-bilinear form on $V^n$.