Results 1 to 3 of 3

Thread: Integral Domain

  1. April 15th 2009, 06:01 PM #1
    chiph588@
    chiph588@ is offline
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163

    Integral Domain

    If R is an integral domain and a, b \in R , show if Ra = Rb then a=ub where u is a unit.
    Follow Math Help Forum on Facebook and Google+
  2. April 15th 2009, 06:20 PM #2
    ThePerfectHacker
    ThePerfectHacker is offline
    Global Moderator ThePerfectHacker's Avatar
    Joined
    Nov 2005
    From
    New York City
    Posts
    10,603
    Quote Originally Posted by chiph588@ View Post
    If R is an integral domain and a, b \in R , show if Ra = Rb then a=ub where u is a unit.
    Since (a) = (b) it means (a)\subseteq (b) \text{ and }(b) \subseteq (a).
    Thus, b|a \text{ and } a|b so a,b are associates and so a=ub.
    Follow Math Help Forum on Facebook and Google+
  3. April 15th 2009, 07:33 PM #3
    NonCommAlg
    NonCommAlg is offline
    MHF Contributor
    Joined
    May 2008
    Posts
    2,296
    Thanks
    6
    Quote Originally Posted by chiph588@ View Post

    If R is an integral domain and a, b \in R , show if Ra = Rb then a=ub where u is a unit.
    to see why we need R to be an integral domain:

    if a=0, then Rb=0 and so b=0. in this case choose u=1. so we may assume that a \neq 0. now we have a \in Ra = Rb. so a=rb, for some r \in R. similarly b=sa, for some s \in R.

    thus a=rb=rsa. hence (1-rs)a=0. since R is an integral domain and a \neq 0, we must have 1-rs=0, i.e. rs=1. so r is a unit and we choose u=r.
    Follow Math Help Forum on Facebook and Google+
« ortogonal projection | Spanning sets? »

Similar Math Help Forum Discussions

  1. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: May 22nd 2011, 11:23 AM
  2. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 12th 2010, 12:47 AM
  3. Integral domain
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 15th 2009, 09:32 AM
  4. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 2nd 2008, 10:54 AM
  5. integral domain?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 23rd 2008, 09:03 AM

Search Tags


/mathhelpforum @mathhelpforum