Definition:Polynomial in Ring Element/Definition 1

Definition
Let $R$ be a commutative ring.

Let $S$ be a subring with unity of $R$.

Let $x\in R$.

A polynomial in $x$ over $S$ is an element $y\in R$ for which there exist: such that:
 * a natural number $n\in \N$
 * $a_0, \ldots, a_n \in S$
 * $y = \displaystyle \sum_{k\mathop = 0}^n a_k x^k$

where:
 * $\sum$ denotes indexed summation
 * $x^k$ denotes the $k$th power of $x$

Also see

 * Equivalence of Definitions of Polynomial in Ring Element