# Definition:Ring of Polynomials in Ring Element

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## Definition

Let $\struct {R, +, \circ}$ be a commutative ring.

Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.

Let $x \in R$.

The subring of $R$ consisting of all the polynomials in $x$ over $D$ is called the **ring of polynomials over $D$** and is denoted $D \sqbrk x$.

## Also known as

Such a ring can also be referred to as a **ring of polynomial forms**, of which definition this is a particular case.

## Also see

- Set of Polynomials over Integral Domain is Subring for a demonstration that $D \sqbrk x$ is indeed a subring of $R$.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64.1$ Polynomial rings over an integral domain