- #1
cosmic_tears
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Hello, and thank you VERY MUCH for reading!
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...
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.
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.
Homework Statement
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...
Homework 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.
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.