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Math Help - Algebra, Properties of a Ring

  1. #1
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    Algebra, Properties of a Ring

    Does anyone know how to prove this:

    Suppose that R is a ring. Show that -(a+b) = (-a)+(-b) for a, b in R.

    Thanks guys.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by suedenation
    Does anyone know how to prove this:

    Suppose that R is a ring. Show that -(a+b) = (-a)+(-b) for a, b in R.

    Thanks guys.
    Actually, it isn't unless the ring is commutative. In general the inverse of (a+b) is (-b)+(-a).

    Calculate the following:

    (a+b)+[(-b)+(-a)]=a+b+(-b)+(-a) by the associative property
    =a+[b+(-b)]+(-a) again by associative
    =a+0+(-a) by definition of inverse
    =a+(-a) by the zero property
    =0 by definition of inverse.

    Thus (a+b)+[(-b)+(-a)] = 0.
    Since the inverse of any element in the ring is unique, (-b)+(-a) = -(a+b).

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark
    Actually, it isn't unless the ring is commutative. In general the inverse of (a+b) is (-b)+(-a).

    Calculate the following:

    (a+b)+[(-b)+(-a)]=a+b+(-b)+(-a) by the associative property
    =a+[b+(-b)]+(-a) again by associative
    =a+0+(-a) by definition of inverse
    =a+(-a) by the zero property
    =0 by definition of inverse.

    Thus (a+b)+[(-b)+(-a)] = 0.
    Since the inverse of any element in the ring is unique, (-b)+(-a) = -(a+b).

    -Dan
    It is commutative, a ring has a property that the binary operation must be commutative. However, a commutative ring is one in such multiplication is also.
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  4. #4
    Forum Admin topsquark's Avatar
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    Pffl! That's three posts in a row I've got at least a detail wrong. I'm slipping!

    Good catch, ThePerfectHacker.

    -Dan
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