I found a problem in this paper, Lemma 2.1.
Let $R$ be a ring and $S\subseteq R$ a multiplicative subset. The set of all $S$-finite right ideals of ring $S$ is a right Oka family.
I've proved this one, but the sentence "Since $R$ is finitely generated it is $S$-finite" make me confuse because it didn't say anything about finiteness of $R$. Why $R$ is finitely generated ?
An ideal $I$ is said to be $S$-finite, if there is a finitely generated ideal $J$ and some $s \in S$ such that $Is \subseteq J \subseteq I$. Now as $S$ is a multiplicative system we should have $1\in S$. Furthermore $R$ is finitely generated as ideal by $R = (1)$. Thus, letting $I=J=R$ and $s=1$, we get that $R$ is $S$-finite.