Definition:Ring of Polynomials in Ring Element

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 $R$.

• Definition:Ring of Polynomial Functions

Sources

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