Currently, I am trying to follow the proof of Proposition 2.5 in this paper. The proposition and proof are shown below, where $S$ is a multiplicatively closed subset of the commutative ring $R$.
I just have one question about this proof. It is the third sentence of the proof. Why does there exist such a $t$? Is $t$ just another name for $1/s$? If this is the case, how we know that $s$ is invertible? From my knowledge, it's not necessary that every element of the multiplicatively closed subset $S$ is invertible.
Thank you!
Best Answer
The fraction $f_2(a_2)/s=0/1$ in the localization $S^{-1}A_3$.
Now recall when two fractions are same in a localization. This mean that there exists $t\in S$ such that $t(f_2(a_2)\cdot 1-0\cdot s)=0$ in $A_3$. There you go!