Results 1 to 2 of 2

Math Help - Group of Units

  1. #1
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722

    Group of Units

    Show that the set of units, U, of R forms a group under multiplication.

    My thoughts... U is a group since if we multiply a unit by a unit we get a unit. (not sure if true and would probably need a proof...)
    Formally, let a,b be units. Then a|1 and b|1. So ab|1, hence ab is a unit. So is a group under multiplication...

    Just a draft of what i think the answer is. Pretty sure there's more to it though. Am i on the right track?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member TheAbstractionist's Avatar
    Joined
    Apr 2009
    Posts
    328
    Thanks
    1
    Quote Originally Posted by Deadstar View Post
    Show that the set of units, U, of R forms a group under multiplication.

    My thoughts... U is a group since if we multiply a unit by a unit we get a unit. (not sure if true and would probably need a proof...)
    Formally, let a,b be units. Then a|1 and b|1. So ab|1, hence ab is a unit. So is a group under multiplication...

    Just a draft of what i think the answer is. Pretty sure there's more to it though. Am i on the right track?
    Hi Deadstar.

    1 is clearly a unit since 1\cdot1=1=1\cdot1.

    If a and b are units, then aa'=1=a'a and bb'=1=b'b for some a',b'\in R. Hence (ab)(b'a')=1=(b'a')(ab) showing that ab is a unit.

    And the above also shows that the multiplicative inverse of a is a'. Hence the set of all units of a ring with unity forms a group under multiplication.

    Notice that I have been careful not to assume that R is a commutative ring since you did not state whether it is or not.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group of Units
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 20th 2011, 06:02 AM
  2. Group of units
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: November 1st 2010, 02:05 AM
  3. Replies: 3
    Last Post: April 21st 2010, 04:53 PM
  4. Replies: 3
    Last Post: November 15th 2008, 05:18 AM
  5. Group of Units (Abelian)
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: April 30th 2008, 09:11 PM

Search Tags


/mathhelpforum @mathhelpforum