Suppose R is any commutative noetherian ring with 1 and R is a direct limit of rings rings $R_{i}$.

Now suppose $I$ is any ideal in $R$. Does there direct system of ideals $ \lbrace I_{i} \rbrace$ of $ \lbrace R_{i} \rbrace$ ( where each $I_{i}$ is an ideal of $R_{i}$) so that $ I = colim I_{i}$ and $R/I = colim_{i} R_{i}/I_{i}$?

I was trying to intersect $I$ with $R_{i}$( in case if each $R_{i} \subseteq R$) but that may not be ideal in $R_{i}$. This is usually a trick in case of modules and submodules but the complexity here is that there are various rings. Any help would be great.