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:

a natural number $n \in \N$

$a_0, \ldots, a_n \in S$

such that:

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

where:

$\ds \sum$ denotes indexed summation

$x^k$ denotes the $k$th power of $x$

Also see

• Equivalence of Definitions of Polynomial in Ring Element

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain