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

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

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

Let $x \in R$.

Let $D \sqbrk x$ denote the ring of polynomials in $x$ over $D$.

Then $D \sqbrk x$ is the smallest subring of $R$ which contains $D$ as a subring and $x$ as an element.