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  1. #1
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    ring questions

    These are my answers to a past exam question, just wondering if someone could please check whether they are right and also whether I have written enough to gain the full marks for each question?

    Thanks in advance!

    Is 3 + 15Z a zero divisor on R? [3 marks]

    Yes because 3.5 = 0.

    What is the inverse of 7 + 15Z in R? [3 marks]

    The inverse is 13 + 15Z because (7 + 15Z)(13 + 15Z) = 1.

    Is 7 + 15Z a zero divisor in R? [3 marks]

    No because there are no such elements that a.7 = 0, where a is in R.

    Is R a field? [2 marks]

    No because not every non-zero element has an inverse in R.

    Find an ideal I of R consisting of 3 elements. [4 marks]

    An ideal of R consisting of 3 elements is {0, 5, 10}.

    What is the number of elements of R/I? [2 marks]

    R/I = 15/{0,5,10} = 15/3 = 5 elements.

    Write down the multiplication table of R/I. [4 marks]

    X 0 1 2 3 4
    0 1 1 2 3 4
    1 0 2 4 1 3
    2 0 2 4 1 3
    4 0 4 3 2 1
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  2. #2
    Lord of certain Rings
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    Congratulations!
    You have done well in all questions except the last one


    I have commented on the answers and given suggestions. The underlined advices are suggestions just for you reference.


    Quote Originally Posted by hunkydory19 View Post
    These are my answers to a past exam question, just wondering if someone could please check whether they are right and also whether I have written enough to gain the full marks for each question?

    Thanks in advance!

    Is 3 + 15Z a zero divisor on R? [3 marks]

    Yes because 3.5 = 0.

    More precisely: (3+15Z)(5+15Z) = 15+15Z = 0 + 15Z


    What is the inverse of 7 + 15Z in R? [3 marks]

    The inverse is 13 + 15Z because (7 + 15Z)(13 + 15Z) = 1.

    Well Done

    Is 7 + 15Z a zero divisor in R? [3 marks]

    No because there are no such elements that a.7 = 0, where a is in R.

    Generall
    y: "If an inverse exists for an element a, then it cannot be a zero divisor". If a.b = 0(with a and b not zero) and a^{-1} exists, then a^{-1}(a.b) = a^{-1}.0 => b = 0. Contradiction.

    Is R a field? [2 marks]
    No because not every non-zero element has an inverse in R.
    Well Done. You should also illustrate a counter example. So you can write "No because not every non-zero element has an inverse in R. For instance, 3 does not have an inverse".

    Find an ideal I of R consisting of 3 elements. [4 marks]

    An ideal of R consisting of 3 elements is {0, 5, 10}.

    What is the number of elements of R/I? [2 marks]

    R/I = 15/{0,5,10} = 15/3 = 5 elements.

    More precisely: [0] = {0,5,10},[1] = {1,6,11}, [2] = {2,7,12}, [3] = {3,8,13},[4] = {4,9,14}.

    Write down the multiplication table of R/I. [4 marks]

    X 0 1 2 3 4
    0 1 1 2 3 4
    1 0 2 4 1 3
    2 0 2 4 1 3
    4 0 4 3 2 1

    Its wrong.The table should look like this:

    X [0] [1] [2] [3] [4]
    [0] 0 0 0 0 0
    [1] 0 1 2 3 4
    [2] 0 2 4 6 8
    [3] 0 3 6 9 12
    [4] 0 4 8 12 1


    But 6 belongs to [1], 8 belong to [3], 12 belongs to [2] and 9 belongs to [4]
    Thus:
    X [0] [1] [2] [3] [4]
    [0] 0 0 0 0 0
    [1] 0 1 2 3 4
    [2] 0 2 4 1 3
    [3] 0 3 1 4 2
    [4] 0 4 3 2 1

    A Note: The point of this exercise was to convince you that R/I is a multiplicative group. Can you see it?
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  3. #3
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    Thank you so much for that Iso, really amazing of you to do that, taught me loads
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