Results 1 to 2 of 2

Math Help - Prove a ring

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    84

    Prove a ring

    Define a new operation of addition in Z by x(+)y = x + y -1 with a new multiplication in Z by x(*)y = x+y-xy. Verify that Z forms a ring with respect to these operations.

    In order to show it is a ring, I must show that R forms an abelian group with respect to addition, R is closed with respect to an associative multiplication, and two distributive laws hold in R.

    How should I start proving those 3 things.

    for abelian group: x + y -1 = y + x -1 and x+y-xy = y+x-yx is this right?

    for associative : (x+y) - xy = x + ( y - xy ) is this right?

    for distributive law: z(x+y-1) = (x+y-1)z and z(x+y-xy) = (x+y-xy)z is this right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by rainyice View Post
    Define a new operation of addition in Z by x(+)y = x + y -1 with a new multiplication in Z by x(*)y = x+y-xy. Verify that Z forms a ring with respect to these operations.

    In order to show it is a ring, I must show that R forms an abelian group with respect to addition, R is closed with respect to an associative multiplication, and two distributive laws hold in R.

    How should I start proving those 3 things.

    for abelian group: x + y -1 = y + x -1 and x+y-xy = y+x-yx is this right?

    for associative : (x+y) - xy = x + ( y - xy ) is this right?

    for distributive law: z(x+y-1) = (x+y-1)z and z(x+y-xy) = (x+y-xy)z is this right?
    To prove that this is an abelian group under + you must also prove that it is a group! You need to find the additive identity, prove that your group is closed (although it is as you cannot add or multiply two integers and not get an integer!) and prove it is associative.

    Currently, you have only shown that addition is commutative (you also proved that multiplication is commutative which do not need to prove). The rest isn't proving anything...

    To prove that addition (I have denoted this by \circ as this is NOT the normal addition) is associative, you need to prove that

    (x\circ y)\circ z = x \circ (y\circ z) where x \circ y = x+y-1.

    Does that make sense? Similarly, multiplication is different in this ring so you should use a different symbol for it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prove that for all a,b in a ring R, (-a)b=-ab=a(-b)
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 6th 2011, 10:26 AM
  2. [SOLVED] Prove the Artinian ring R is a division ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 8th 2011, 04:53 AM
  3. example of prime ring and semiprime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 27th 2011, 06:23 PM
  4. Ideals of ring and isomorphic ring :)
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 24th 2009, 04:23 AM
  5. prove trig identity by using commutative ring theory
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 21st 2007, 02:31 PM

Search Tags


/mathhelpforum @mathhelpforum