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Math Help - subring of the integer set

  1. #1
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    subring of the integer set

    Why isn't Z_2 = \{0, 1\} a subring of the integer set Z?
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    Quote Originally Posted by dori1123 View Post
    Why isn't Z_2 = \{0, 1\} a subring of the integer set Z?
    Because it is not even a subset !
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    why not? Z_2 is contained in Z, isn't it?
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    Quote Originally Posted by dori1123 View Post
    why not? Z_2 is contained in Z, isn't it?
    Ah! But remember it is not really \{ 0,1\} it is really \{ [0]_2,[1]_2\}.

    Where, [0]_2 = \{ 0, \pm 2, \pm 4, ... \} and [1]_2 = \{ \pm 1, \pm 3, \pm 5, ... \}.

    Therefore, \mathbb{Z}_2 is a group modulo 2 under addition.
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    Maybe she has mistaken it with 2\mathbb{Z} which are the even integers and indeed is a subring of the integers, and so are n\mathbb{Z}
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    That depends on what is a ring: if you make the difference between rings and pseudo-rings, the only subring of \mathbb{Z} is \mathbb{Z} itself.
    Last edited by clic-clac; November 27th 2008 at 09:48 AM. Reason: better
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    Please, donīt make things more complicated.

    A ring is a ring is a ring....
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    Well "my" definition of a ring A states that A has a multiplicative identity...
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    Quote Originally Posted by dori1123 View Post
    Why isn't Z_2 = \{0, 1\} a subring of the integer set Z?
    Another way of putting it is that the addition operation is different. In \mathbb{Z}, 1+1=2. But in \mathbb{Z}_2, 1+1=0.

    Part of the definition of a subring is that it must inherit the algebraic operations of the full ring.
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  10. #10
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    I donīt think it was necessary to lower my reputation for my comment; obviously we are all talking about rings with unity, so my comment about the subrings of \mathbb{Z} is correct in that context.
    Bringing to the table the issue of pseudo-rings (or rngs) had nothing to do with the question, must insist...
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  11. #11
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    Inti, maybe I don't understand, but what's the unity in n\mathbb{Z}?

    ps. I didn't lower your reputation
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