Results 1 to 4 of 4

Math Help - Algebra, Properties of a Ring

  1. #1
    Junior Member
    Joined
    Feb 2006
    From
    Canada
    Posts
    45

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,899
    Thanks
    329
    Awards
    1
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,899
    Thanks
    329
    Awards
    1
    Pffl! That's three posts in a row I've got at least a detail wrong. I'm slipping!

    Good catch, ThePerfectHacker.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. properties of radical ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 12th 2011, 12:46 PM
  2. Replies: 0
    Last Post: April 23rd 2010, 11:37 PM
  3. Ring Properties
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 20th 2009, 02:37 PM
  4. linear Algebra: Properties of Determinants
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 18th 2007, 12:00 AM
  5. Abstract Algebra Reflexive Properties
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: October 30th 2006, 08:59 AM

Search Tags


/mathhelpforum @mathhelpforum