# Definition:Polynomial in Ring Element

## Definition

Let $R$ be a commutative ring.

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

Let $x\in R$.

### Definition 1

A polynomial in $x$ over $S$ is an element $y \in R$ for which there exist:

a natural number $n \in \N$
$a_0, \ldots, a_n \in S$

such that:

$y = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where:

$\displaystyle \sum$ denotes indexed summation
$x^k$ denotes the $k$th power of $x$

### Definition 2

Let $S[X]$ be the polynomial ring in one variable over $S$.

A polynomial in $x$ over $S$ is an element that is in the image of the evaluation homomorphism $S[X]\to R$ at $x$.