A quick question on the subring generated by a finite set

Question

Let $R$ be a commutative unital ring and let $r_1,\ldots,r_n \in R$.

Let $S$ be the unital subring of $R$ generated by $r_1,\ldots,r_n$.

Let $\varphi:\mathbb{Z}[X_1,\ldots,X_n]\to R$ be the unique homomorphism sending $1\mapsto 1_R$ and each $X_i\mapsto r_i$.

Am I right in thinking that $S=\text{im}(\varphi)$?

Many thanks!

Answer 1

Yes. It is clear that $\text{im}(\phi) \subseteq S$. On the other hand, $\text{im}(\phi)$ is a subring of $R$ that contains all the $r_i$'s, and since $S$by definition is the smallest subring of $R$ containing the $r_i$'s, we must have $S \subseteq \text{im}(\phi)$. Hence, $S = \text{im}(\phi)$.