Results 1 to 2 of 2

Math Help - Commutative ring testing

  1. #1
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Commutative ring testing

    Consider the set
    Z[\sqrt{3}]=(a+b\sqrt{3} | a, b \in{Z})

    a) Show that Z[\sqrt{3}] is a commutative ring
    b) Show that if a+b\sqrt{3}=c+d\sqrt{3} then a=b and c=d

    Answers
    a) Is it sufficient to show that Z[\sqrt{3}] satisfies additive commutative, associativity, identity and inverse and satisfies multiplicative associativity, commutativity and identity?
    In that case

    Noting that Zis a commutative ring

    Additive commutativity
    (a+b\sqrt{3})+(c+d\sqrt{3})=((a+c)+(b+d)\sqrt{3})
    (c+d\sqrt{3})+(a+b\sqrt{3})=((c+a)+(d+b)\sqrt{3})

    The above statements are equal to each other, then Z[\sqrt{3}] satisfies this property.

    Multiplicative commutativty
    (a+b\sqrt{3})*(c+d\sqrt{3})=(ac+ad\sqrt{3}+bc\sqrt  {3}+3bd)
    (c+d\sqrt{3})*(a+b\sqrt{3})=(ca+cb\sqrt{3}+da\sqrt  {3{+3db)

    Above 2 expressions are equal to each other, then Z[\sqrt{3}] satisfies this property.

    Do I continue in this vein with the other properties to prove that Z[\sqrt{3}] is a commutative ring?

    Part B
    a+b\sqrt{3}=c+d\sqrt{3}

    Is it sufficient to show that
    (a-c)+(b-d)\sqrt{3}=0+0\sqrt{3}
    This a-c=0 and b-d=0
    Hence a=c and b=d
    Will this suffice?

    Thanks for the responses in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2007
    Posts
    90
    That is what I would do. It's a boring exercise to get you learning the ring axioms.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 22nd 2011, 11:38 AM
  2. Commutative ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 8th 2010, 07:05 AM
  3. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 24th 2009, 09:54 AM
  4. Non-commutative ring
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: March 22nd 2009, 03:12 AM
  5. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 20th 2008, 11:51 AM

Search Tags


/mathhelpforum @mathhelpforum