What are the discrete subrings of the real set?

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In summary, discrete rings are a structure where a subset of a ring is discrete if it carries the discrete topology as a subspace. Z is the only discrete subring of R.
  • #1
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Homework Statement


Problem from Artin's Algebra, find all discrete subrings of the real set.

The Attempt at a Solution



Clearly, Zn = {...,-2n,-n,0,n,...} is a portion. But having trouble proving that this forms *all* of the discrete subgroups.
 
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  • #2
Can you first tell me what a discrete ring is? Is it just the following: [tex]S\subseteq \mathbb{R}[/tex] is a discrete ring if

[tex]\exists \epsilon >0:~\forall x\in S\setminus \{0\}:~|x|>\epsilon[/tex]

I'll assume that this is your definition of discrete...

Now, of course nZ is a discrete subgroup of R, but it is not a discrete subring (since 1 is not in nZ). The only n that gives a subring is n=1. Thus we must prove that Z is the only discrete subring of R.

Let's start with an example: if 1/2 is an element of our discrete subring S, can you find arbitrary small elements in S? (hint: multiply 1/2 by itself)
 
  • #3
micromass said:
Can you first tell me what a discrete ring is? Is it just the following: [tex]S\subseteq \mathbb{R}[/tex] is a discrete ring if

[tex]\exists \epsilon >0:~\forall x\in S\setminus \{0\}:~|x|>\epsilon[/tex]

I'll assume that this is your definition of discrete...

Now, of course nZ is a discrete subgroup of R, but it is not a discrete subring (since 1 is not in nZ). The only n that gives a subring is n=1. Thus we must prove that Z is the only discrete subring of R.

Let's start with an example: if 1/2 is an element of our discrete subring S, can you find arbitrary small elements in S? (hint: multiply 1/2 by itself)

Yes, sorry, you're right. I learned originally that rings do not necessarily contain unity, but I see Artin defined them such that they do. Nice proof.

I took discrete to mean:
[tex]\exists \epsilon >0 : \forall (x,y \in S : x \not= y) \; |x-y|>\epsilon[/tex]
Well, if you can read this, I'm not so great at tex. But I don't think this point matters. Wikipedia says this is supposed to correspond to the discrete topology somehow... are these equivalent formulations?

If we relaxed this requirement of unity for the subrings, would the set of all Zn compose all of the discrete subrings of [tex]\Re[/tex]?
 
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  • #4
Ah, yes. It does correspond to the discrete topology. In fact, a subset S of R is discrete (in your sense) iff it carries the discrete topology as subspace of R.

And it is indeed true that if a ring doesn't need to contain a unity, that all the nZ will be discrete subrings of R. And they will be the only discrete subrings of R...
 

1. What is a discrete subring of R?

A discrete subring of R is a subset of the real numbers that forms a ring, meaning it is closed under addition, subtraction, and multiplication, and contains the identity element for each operation. Additionally, the subset must be discrete, meaning it has no accumulation points.

2. How do you determine if a subset of R is a discrete subring?

To determine if a subset of R is a discrete subring, you must first check if it is closed under addition, subtraction, and multiplication. Then, you must check if it contains the identity element for each operation. Finally, you must verify that the subset does not have any accumulation points.

3. Can a discrete subring of R be infinite?

Yes, a discrete subring of R can be infinite. As long as the subset satisfies the requirements of being closed under addition, subtraction, and multiplication, and does not have any accumulation points, it can be infinite.

4. How are discrete subrings of R different from continuous subrings?

The main difference between discrete subrings of R and continuous subrings is that discrete subrings do not have any accumulation points, while continuous subrings can have accumulation points. This means that the elements in a discrete subring are separated by positive distances, while the elements in a continuous subring can be arbitrarily close to each other.

5. What are some examples of discrete subrings of R?

Some examples of discrete subrings of R include the set of integers (Z), the set of rational numbers (Q), and the set of algebraic numbers (numbers that can be expressed as roots of polynomials with integer coefficients).

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