Examples for ideals that can not be finitely generated In $\mathbb{Z}$ every ideal can be generated by $1$ element, so $\mathbb {Z}$  is a principal ideal domain.
I'm looking for specific examples (rings) for which


*

*every ideal can be generated by at most $2$ elements

*every ideal can be generated by at most $3$ elements

*...

*every ideal can be generated by a finite set of elements

*there are ideals that cannot be generated by a finite set of elements
Does this question/search make sense at all?
 A: Rings for which every ideal is generated by a set of $n$ elements or less are said to have the n-generator property. I'm not too sure about the specifics of examples for each particular $n$. There is probably a family of examples out there somewhere.
Dedekind domains are prominent examples of rings with the $2$-generator property. In fact, they are informally said to have the property that their ideals are "$1\frac12$-generated."  I can't remember exactly the reason for this: I thought I remembered it had to do with being able to choose one of the elements. 1
The commutative rings for which the ideals are all finitely generated are exactly the Noetherian rings.
In light of the last answer, we can say that a (commutative) ring has a non-finintely generated ideal iff it is not Noetherian, so any non-Noetherian ring will do.
What I've said for the last two questions holds equally well for noncommutative rings, but I've just been keeping everything in the straightforward context of commutative rings with identity.
