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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 anidealin 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 isnotin 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.

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

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