Results 1 to 2 of 2

Math Help - Boolean ring, induction

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    23

    Boolean ring, induction

    A commutative ring A is Boolean if x^2  = x for all x \in A.
    In a Boolean ring A, show that every finitely generated ideal in A is principal.

    I am pretty sure this proof is by induction because: (x,y)=(x-xy+y) The inclusion (\supseteq) is clear, and the identities x(x-xy+y)=x and y(x-xy+y)=y give (\subseteq). How do I prove this by induction?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by zelda2139 View Post
    A commutative ring A is Boolean if x^2 = x for all x \in A.
    In a Boolean ring A, show that every finitely generated ideal in A is principal.

    I am pretty sure this proof is by induction because: (x,y)=(x-xy+y) The inclusion (\supseteq) is clear, and the identities x(x-xy+y)=x and y(x-xy+y)=y give (\subseteq). How do I prove this by induction?
    first of all you don't need to add "commutativity" in the definition of a Boolean ring because it's a result of the condition x^2=x, \ \forall x \in A. to complete your induction just note that

    I=<x_1, \cdots , x_{n-1}, x_n>=Ax_1 + \cdots + Ax_{n-1}+Ax_n. now if J=<x_1, \cdots, x_{n-1}>=<x>, then I=J+Ax_n=<x,x_n>=<x-xx_n+x_n>.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prove the Artinian ring R is a division ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 8th 2011, 03:53 AM
  2. example of prime ring and semiprime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 27th 2011, 05:23 PM
  3. Boolean commutative ring
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 7th 2011, 11:01 PM
  4. Boolean Ring
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 18th 2009, 01:19 PM
  5. Boolean ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 3rd 2009, 05:49 PM

Search Tags


/mathhelpforum @mathhelpforum