Subring of Polynomials over Integral Domain is Smallest Subring containing Element and Domain

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

Let $\left({D, +, \circ}\right)$ be an integral domain such that $D$ is a subring of $R$.

Let $x \in R$.

Let $D \left[{x}\right]$ denote the subring of polynomials in $x$ over $D$.

Then $D \left[{x}\right]$ is the smallest subring of $R$ which contains $D$ as a subring and $x$ as an element.