0
$\begingroup$

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!

$\endgroup$
0
$\begingroup$

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, $S = \text{im}(\phi)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.