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Math Help - Units and Zero Divisors

  1. #1
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    Units and Zero Divisors

    The question posed, is Prove a matrix A is a zero divisor if and only if it A is singular.
    I have proved this statement.

    The question is this true for all rings?
    So i guess the question is for any ring R, An element a is a zero divisor if and only if it is not a unit.

    ===> Assume by contradiction a is a unit,
    But a is a zero divisor, so there exist a b belong to R b not equal to 0 such that ab = 0
    but we assume a is a unit so a^-1(ab) = (a^-1)0 = 0 which implies b = 0 which is a contradiction.

    <== I having trouble with the reverse. We assume that A is not a unit. I am not sure where to go can someone help?

    I almost positive that is it is true that if an element is unit then it is not a zero and likewise if an element is a 0 divisor then it is not a unit.

    I am having problems showing if an element is a zero divisor then it cant be a unit
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  2. #2
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    Quote Originally Posted by john890 View Post
    The question posed, is Prove a matrix A is a zero divisor if and only if it A is singular.
    I have proved this statement.

    The question is this true for all rings?
    So i guess the question is for any ring R, An element a is a zero divisor if and only if it is not a unit.

    ===> Assume by contradiction a is a unit,
    But a is a zero divisor, so there exist a b belong to R b not equal to 0 such that ab = 0
    but we assume a is a unit so a^-1(ab) = (a^-1)0 = 0 which implies b = 0 which is a contradiction.

    <== I having trouble with the reverse. We assume that A is not a unit. I am not sure where to go can someone help?

    I almost positive that is it is true that if an element is unit then it is not a zero and likewise if an element is a 0 divisor then it is not a unit.

    I am having problems showing if an element is a zero divisor then it cant be a unit


    What you want to prove is false. Example, in \mathbb{Z} , 2 is not a unit but it isn't azero divisor. Try the same with

    any integral domain which isn't a field.

    Tonio
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