Let $S$ be a multiplicatively closed subset of a commutative ring $R$. Then is it true that the localization $R_S=\{r/s:r\in R, s\in S\}$ a sub-ring of $R$?
I think it is true, because $r/s=rs^{-1}$ and ring is closed under multiplication. Am I right? Sorry, I know this is a trivial question, but I am just not sure. Thank you.
On the contrary, there is a (canonical) homomorphism $\varphi: R \to S^{-1}R$, such that $\varphi(x)=\dfrac x1$, which is injective if $S$ contains no zero-divisors. The best known example is the field of fractions of an integral domain, which certainly is not a subring of the domain (unless the latter be already a field, of course).