Polynomial ring as direct sum of modules

Question

I'm aware that this topic has been posted previously. However, I'm not wanting to prove anything. I just want to understand, somewhat intuitively, why $R[x]$ as a module over $R$ is given by $$R[x] =\bigoplus_{i=1}^{\infty} R?$$

Answer 1

What's an element of $R[x]$? It's a finite sum that looks like this: $$a_0+a_1x + \cdots + a_n x^n,$$ where the $a_i \in R$. If we forget about the variable $x$ and note that all that really matters in this description is the coefficients $a_i$ and the order they appear in, we realize that this corresponds in a bijective fashion to an ordered tuple (with finitely many nonzero elements)$$(a_0,a_1,\ldots, a_n,0,0,\ldots) \in \bigoplus_{i \in \mathbb{N}} R.$$

As this correspondence preserves the module operations, it is an isomorphism.