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Thread: Subring

  1. #1
    Forum Admin topsquark's Avatar
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    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
    Last edited by topsquark; Aug 8th 2018 at 07:30 PM.
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    Re: Subring

    Quote Originally Posted by topsquark View Post
    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.
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    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.
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    Forum Admin topsquark's Avatar
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    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
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    Re: Subring

    Quote Originally Posted by topsquark View Post
    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.
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    Re: Subring

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

    Thanks!

    -Dan
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