\begin{definition}[Definition:Coefficient of Polynomial]
Let $R$ be a commutative ring with unity.
Let $P \in R[x]$ be a polynomial over $R$.
By Monomials Form Basis of Polynomial Ring, the set $\{x^k : k \in \N\}$ is a basis of $R[x]$.
By Equality of Monomials of Polynomial Ring, all $x^k$ are distinct.
The '''coefficient''' of $x^k$ in $P$, or the '''$k$th coefficient''' of $P$, is the coefficient of $x^k$ in $P$.
{{DefinitionWanted|define coefficient wrt basis of module}}
\end{definition}