Definition:Ring of Polynomials in Ring Element

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

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

The subring $R$ consisting of all the polynomials in $x$ over $D$ is called the ring of polynomials over $D$ and is denoted $D \left[{x}\right]$.

Also see

 * Set of Polynomials over Integral Domain is Subring for a demonstration that $D \left[{x}\right]$ is indeed a subring of $R$.