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Math Help - unity and units

  1. #1
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    unity and units

    Let R be a nonzero commutative ring, and let T be a nonempty subset of R closed under multiplication and containing neither 0 nor divisors of 0.

    Show:
    a) Q(R,T) has a unity even if R doesn't
    b) In Q(R,T) every nonzero element of T is a unit.

    c) how many elements are there in the ring Q(Z4, {1,3})
    d) describe the ring Q(3Z, {6^N | N is in Z+}) by describing a subring of R to which it is isomorphic.

    Thanks
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  2. #2
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    Quote Originally Posted by lttlbbygurl View Post
    Let R be a nonzero commutative ring, and let T be a nonempty subset of R closed under multiplication and containing neither 0 nor divisors of 0.

    Show:
    a) Q(R,T) has a unity even if R doesn't
    i've never seen this notation before! by Q(R,T) do you mean T^{-1}R, the localization of R at T? in this case just choose any elements of t \in T. then \frac{t}{t} is the unity of Q(R,T).

    b) In Q(R,T) every nonzero element of T is a unit.
    we, as you mentioned yourself, assume that 0 \notin T. so saying something like "every non-zero element of T" is not necessarily! if R has no unity, the inverse of an element

    t \in T in Q(R,T) is \frac{t}{t^2}. if R has a unity, then the inverse of t in Q(R,T) is simply \frac{1}{t}.


    c) how many elements are there in the ring Q(Z4, {1,3})
    4 elements. note that in Q(\mathbb{Z}_4, \{1,3 \}) we have \frac{1}{3}=3 and \frac{2}{3}=2. in fact Q(\mathbb{Z}_4, \{1,3 \})=\mathbb{Z}_4.


    d) describe the ring Q(3Z, {6^N | N is in Z+}) by describing a subring of R to which it is isomorphic.

    Thanks
    i'm not sure what this part is trying to say! is it claiming that Q(3\mathbb{Z}, \{6^n: \ n \in \mathbb{Z}_+ \}) is isomorphic to a subring of 3\mathbb{Z}??
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post

    i'm not sure what this part is trying to say! is it claiming that Q(3\mathbb{Z}, \{6^n: \ n \in \mathbb{Z}_+ \}) is isomorphic to a subring of 3\mathbb{Z}??

    Yes, sorry for the confusion
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