# Equivalence of Definitions of Polynomial in Ring Element

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## Theorem

Let $R$ be a commutative ring.

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

Let $x\in R$.

The following definitions of the concept of **Polynomial in Ring Element** are equivalent:

### 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:

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

where:

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

### Definition 2

Let $S \sqbrk X$ 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 $S \sqbrk X \to R$ at $x$.

## Proof

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