What are the discrete subrings of the real set?

[PhDorBust]

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.

 

[micromass]

Can you first tell me what a discrete ring is? Is it just the following: $$S\subseteq \mathbb{R}$$ is a discrete ring if

$$\exists \epsilon >0:~\forall x\in S\setminus \{0\}:~|x|>\epsilon$$

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)

 

[PhDorBust]

Can you first tell me what a discrete ring is? Is it just the following: $$S\subseteq \mathbb{R}$$ is a discrete ring if

$$\exists \epsilon >0:~\forall x\in S\setminus \{0\}:~|x|>\epsilon$$

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:
$$\exists \epsilon >0 : \forall (x,y \in S : x \not= y) \; |x-y|>\epsilon$$
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 $$\Re$$?

 

[micromass]

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...