If I take $3$ random polynomials $φ,ψ,ω$ on a ring $R[x]$, I'm trying to prove associativity which is very obvious. But I have trouble on the algebra part with the sums. I know that given $2$ polynomials $φ,ψ$ we have $φ*ψ=κ$, where $$\begin{align} φ &= (a(0),a(1),\dots), \qquad (i=0,1,2, \dots), \\ ψ &= (b(0),b(1),\dots), \qquad (j=0,1,2, \dots), \\ κ &= (d(0),d(1),\dots), \qquad (m=0,1,2, \dots), \\ d(m) &= \sum_{i+j=m} a(i)*b(j). \end{align}$$
Please help if you can.
Hints:
Did you try a little induction of the sum of the degrees? That can help quite a bit...
$$\sum_{i=0}^na_ix^i\sum_{k=0}^mb_kx^k=a_0b_0+x^2\sum_{i=1}^na_ix^{i-1}\sum_{k=1}^mb_kx^{k-1}$$
But then both summands on the right are polynomials the sum of which degrees is less than the original ones'...