1. ## Subring

I'm just starting in on rings and I have a question that I simply have no clue on.

Given the ring of functions $\displaystyle f:~ [0,~1] \to \mathbb{R}$, determine if the restriction on f such that $\displaystyle \lim _{x \to 1^-} f(x) = 0,~x \in [0,~1]$ forms a subring of f.
What the heck does this restriction have to do with a ring structure? I'm totally lost.

-Dan

2. ## Re: Subring

Originally Posted by topsquark
I'm just starting in on rings and I have a question that I simply have no clue on.
Given the ring of functions f: [0, 1]→R, determine if the restriction on f such that limx→1−f(x)=0, x∈[0, 1] forms a subring of f.
What the heck does this restriction have to do with a ring structure? I'm totally lost.
Do you understand that if $\mathcal{R}$ is a ring and $\mathcal{S}\subseteq\mathcal{R}$ having the properties that:
$i)~\mathcal{S}\ne\emptyset$
$ii)~\forall a~\&~b\in\mathcal{S}[ab\in\mathcal{S}~\&~a-b\in\mathcal{S}]~$
Then $\mathcal{S}$ is a subring.

3. ## Re: Subring

The ring itself consists of all functions from [0, 1] to R. The question the asks if the subset of all such functions that satisfy that restriction forms a sub-ring. Basically, you need to determine if that subset is closed under addition, negation, and multiplication.

4. ## Re: Subring

Okay, I can prove that the example I mentioned above is a ring. This is one of a six part problem (not related to each other) that gives the ring of functions $\displaystyle f: [0,~1] \to \mathbb{R}$, the purpose of which seems to be listing different kinds of subrings. The reason I was thinking that the one I posted may not be a subring is because the restriction $\displaystyle \lim_{x \to 1^-} f(x) = 0$ seems so trivial that I was thinking there was some kind of "trick" to it. Thinking too hard about it, I guess.

However there is still a potential problem to my mind. All of these problems deal with subrings that map [0, 1] to $\displaystyle \mathbb{R}$, where we have + and $\displaystyle \cdot$ defined as usual for real numbers. I can't seem to find a set of functions mapping to $\displaystyle \mathbb{R}$ that wouldn't be a subring. And that makes no sense to me.

-Dan

5. ## Re: Subring

Originally Posted by topsquark
Okay, I can prove that the example I mentioned above is a ring. This is one of a six part problem (not related to each other) that gives the ring of functions $\displaystyle f: [0,~1] \to \mathbb{R}$, the purpose of which seems to be listing different kinds of subrings. The reason I was thinking that the one I posted may not be a subring is because the restriction $\displaystyle \lim_{x \to 1^-} f(x) = 0$ seems so trivial that I was thinking there was some kind of "trick" to it. Thinking too hard about it, I guess.

However there is still a potential problem to my mind. All of these problems deal with subrings that map [0, 1] to $\displaystyle \mathbb{R}$, where we have + and $\displaystyle \cdot$ defined as usual for real numbers. I can't seem to find a set of functions mapping to $\displaystyle \mathbb{R}$ that wouldn't be a subring. And that makes no sense to me.

-Dan
Here is a good reference.

6. ## Re: Subring

Originally Posted by Plato
Sounds good. I'm about to order a batch of books anyway. This sounds like a good addition to my list.

Thanks!

-Dan