Definition:Polynomial in Ring Element

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring.

Let $\left({S, +, \circ}\right)$ 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: such that:
 * a natural number $n\in \N$
 * $a_0, \ldots, a_n \in S$
 * $y = \displaystyle \sum_{k\mathop = 0}^n a_k \circ x^k$

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

Definition 2
Let $R[X]$ be the polynomiial ring in one variable over $R$.

A polynomial in $x$ over $S$ is an element that is in the image of the evaluation homomorphism at $x$.

Polynomial over Integral Domain
An important special case is when $S$ is an integral domain:

Also see

 * Definition:Polynomial
 * Definition:Polynomial Function
 * Definition:Polynomial Equation