\begin{definition}[Definition:Multiplication of Polynomials]
Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +, \circ}$ be a subring of $R$.
Let $x \in R$.
Let:
:$\ds f = \sum_{j \mathop = 0}^n a_j x^j$
:$\ds g = \sum_{k \mathop = 0}^n b_k x^k$
be polynomials in $x$ over $S$ such that $a_n \ne 0$ and $b_m \ne 0$.
{{questionable|this does not need to be defined; it's just the product of ring elements}}
The '''product of $f$ and $g$''' is defined as:
:$\ds f g := \sum_{l \mathop = 0}^{m + n} c_l x^l$
where:
:$\ds \forall l \in \set {0, 1, \ldots, m + n}:  c_l = \sum_{\substack {j \mathop + k \mathop = l \\ j, k \mathop \in \Z} } a_j b_k$
\end{definition}