Definition:Polynomial in Ring Element/Definition 2
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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 $x$.