Homework Help: Ring theory - characterizing ideals in a ring.

1. Dec 11, 2008

cosmic_tears

Hello, and thank you VERY MUCH for reading!

1. The problem statement, all variables and given/known data
Let p be a prime number.
Let R= Z(p) be the ring defined as followed:
Z(p) = {x/y : gcd(y,p)=1} (notice that it's not the ring {0,1,...,p-1}!)
I need to characterize all the ideals in this ring, and all of it's quotient rings...

2. Relevant equations
Well, not exactly equations, but just a few defintions:
I is an ideal in R if:
1) it is a subgroup of R under addition.
2) for every a in I and r in R, a*r is in I, and r*a is in I.

3. The attempt at a solution
I already proved Z(p) is a ring (I needed to do so before this question).
I also noticed that an element x/y is invertible if and only if x is not in pZ (meaning, if and only if gcd(x,p)=1).
I know that if an Ideal cosist an invertible element then it is all of R, so I'm seeking for ideals that consist of elements x/y such that gcd(x,p)=1. However, I cannot see how to find how many ideals of this type there are, and more over - how to show that there are no other types of ideals... :-\
I'll think of quotient rings after I find the ideals...

That's it. I really appreciate the fact that you are reading this, and any response is welcomed!
Thanks, bless you, you are a great help!
Tomer.

2. Dec 13, 2008

PatF

This is, I think, called the localization of the integers at the prime p. It really consists of all rational numbers whose denominator is not divisible by p.

So, what does an element look like when it is not invertible? (I think you already know about them.) Pick a prime, say 5, and look at the elements of Z(5). Write down a couple of the non invertible elements. Find relations among them.

( I would write more but that would just hand you the answer and you should think about it because it is not very difficult.)

3. Dec 14, 2008

cosmic_tears

Thanks, PatF, I realized all ideals are of the form (p^k) for a whole k.