Explanation

If V is a subset of R the ideal generated by V could also be defined as the intersection of all ideals containing V.

To see that the ideal defined in the theorem is exactly that, note that, if I is an ideal containing V, then I contains (V). This implies that V is contained in the intersection of all ideals containing V.

On the other hand, the ideal defined in the theorem clearly contains V and so coincides with the intersection.