Evaluation and roots of a polynomial in a commutative target ring

Let $\varphi\colon R\to S$ be a unital ring homomorphism between commutative rings (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$), let $s\in S$, and let $f=\sum_i a_i x^i\in R[x]$. The value of $f$ at $s$ along $\varphi$ is

$f_\varphi(s):=\sum_i\varphi(a_i)s^i\in S.$

The sum is finite because the coefficient sequence of $f$ has finite support (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). When $R$ is identified with a subring of $S$, the inclusion is understood and the subscript is omitted. An element $s\in S$ is a root or zero of $f$ when $f_\varphi(s)=0_S$.

Evaluation produces an element of the target ring from a formal polynomial. It does not identify the polynomial with the function that it induces.