Definition:Polynomial in Ring Element/Definition 2

Definition
Let $R$ be a commutative ring.

Let $S$ be a subring with unity of $R$.

Let $x \in R$.

Let $S \sqbrk X$ be the polynomial ring in one variable over $S$.

A polynomial in $x$ over $S$ is an element that is in the image of the evaluation homomorphism $S \sqbrk X \to R$ at $x$.

Also see

 * Equivalence of Definitions of Polynomial in Ring Element