# Distint ideals of a ring

1. Dec 8, 2011

### me@math

How do you find all the distint ideals of any ring? I am able to find may ideals but how do you prove that there are no more ideals.
Eg Let R = Z[1/n] = {x/n^i | x $\in$ Z, n is a natural number}

I can see that x/n is an ideal for every x $\in$ Z.

Is that right?

2. Dec 8, 2011

### alexfloo

I'm not sure what you're asking. x/n appears to be an element, and an ideal is a subring.

Although I'm not sure, I'll try to answer something that could be want you want:

The integers has a special structure called that of a Principal Ideal Domain. A principal ideal is the ideal generated by a single element: that is an element together with all of its multiples. In a Principal Ideal Domain, every ideal is a principal ideal, so it's easy to identify them all.

For other rings, however, there is not necessarily a general rule like that. It's not obvious to me whether Z adjoined with a reciprocal integer (Z[1/n]) is a PID, though it may be.