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Math Help - Group of units

  1. #1
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    Group of units

    What is the group of units of Z, Q, R[x]???

    I do not know where to start... any hints or clues...

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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
    What is the group of units of Z, Q, R[x]???

    I do not know where to start... any hints or clues...


    In each ring ask yourself: what is the group of elements that multiplied by some other element gives us 1? It's easy.

    Tonio
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  3. #3
    Senior Member Dinkydoe's Avatar
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    Are you serious? Because this is more or less a trivial Question.

    If you know what units are, then it's not so hard to look for them...
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  4. #4
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    From what i read on units...
    the units for Z will be 1 and -1, for Q it will be Q excluding 0... But i dont get R[x], will the units here be R[x] excluding R[0] ?
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  5. #5
    Senior Member Dinkydoe's Avatar
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    No, actually a polynomial f \in R[X] is not invertible if \mathrm{deg}(f) \geq 1. You need to show this. But then you're left with units of R\subset R[X], the invertible polynomials of degree zero, wich is R^*

    So, what is the group units of R?
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  6. #6
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    So Are units just a group of invertible elements... If i wanted the group units of 2 by 2 real matrices will it be all 2 by 2 real matrices where the determinant is not equal to 0... Whats the purpose for finding such groups?
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  7. #7
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    Yes since determinant not 0 means it is invertible. I'm not sure about the purpose... but it can be useful for finding the zero divisors. If you have the list of all units, every nonzero element not in the list is a zero divisor. Because the zero divisors are precisely the nonzero nonunits.
    Last edited by demode; October 30th 2010 at 06:40 PM.
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  8. #8
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Dreamer78692 View Post
    So Are units just a group of invertible elements... If i wanted the group units of 2 by 2 real matrices will it be all 2 by 2 real matrices where the determinant is not equal to 0... Whats the purpose for finding such groups?
    This is maths! There doesn't have to be a purpose!

    The group of units, however, is interesting. It can be used, for example, to find out if two rings are isomorphic or not (none of the three rings you gave are isomorphic as they have non-isomorphic groups of units).

    What is also interesting is this:

    A group is called Linear if it can be found as a subgroup of some general linear group. For example, all finite groups are linear groups. However, not all groups are linear groups! Now, there are rings called PI-rings, and if G is a linear group then G occurs as a group of units of a PI-rings (it is called a PI-group). Not all PI-groups are Linear groups.

    That is, \textnormal{Linear group }\subsetneq\textnormal{ PI-group}.
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