Example

Let a R and put V = {a}. The ideal of the theorem is the set of all multiples of a:

I = {ra | r R}.

In the cases R = Z and R = Q[X], these are exactly the elements equivalent to 0 modulo a. We shall see shortly that this is no coincidence.

Notation: Ra or (a), as usual for, e.g., R = Z and Q[X].