Consider an ideal $I=\langle f\_1,f\_2,\ldots,f\_s\rangle$ in the polynomial ring $\mathbb{Q}[x\_1,x\_2,\ldots,x\_n].$ Build the following set 
$$
\{ g\_1 f\_1+g\_1 f\_2+\cdots+g\_n f\_n \},
$$
where $g\_i$ belongs to the field of fractions $\mathbb{Q}(x\_1,x\_2,\ldots,x\_n)$ and denominators of all $g\_i$ does not belong to the ideal $I.$ Is there a special name for this set?



What is correct name of the following  construction?
This is the image of $I$ in the localization $\mathbb Q[x\_1,x\_2,\dots,x\_n]\_{I}$.


There is an issue here though. If $I$ is not a prime ideal, then its complement is not multiplicatively closed, and therefore not a good set to invert in a localization. If $I$ is not a prime ideal, then there are some $f, g$ in $I^c$ such that $fg\in I$. Thus, e.g., $\frac{1}{f}\cdot\frac{1}{g}$ becomes a non-allowed coefficient, while $\frac{1}{f}$ and $\frac{1}{g}$ both are.


The set you describe can still be defined, obviously, but it will lack interesting structure. If $I$ is a prime ideal, then your set is the ideal in the localization ring that I describe above.


