1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Units of finite rings

  1. Dec 12, 2008 #1
    1. The problem statement, all variables and given/known data
    Let q be the number of units in finite ring R. Show that for all a in R, if a is a unit in R then [tex]a^q = 1[/tex].

    Is there a way to solve this without using group theory? All I can seem to find information on is when a and m are relatively prime then [tex]a^{\phi (m)} = 1 (mod \, m)[/tex], which I'd like to prove using the problem I can't solve.

    2. Relevant equations

    3. The attempt at a solution

    I really haven't been able to get anywhere on this. Are there certain patterns that finite rings always follow, that I can exploit?

  2. jcsd
  3. Dec 12, 2008 #2
    If q is the number of invertibles in the ring R, then it means that the order of V is q. Where V would be the group of invertibles in that ring. That is we know that V( the set of all invertibles in a ring is a group in itself). Now it is clear that if a is invertible(unit) then it belongs to V. So, by lagrange theorem we have the desired result that a^q=1. where 1 is the unity of the ring R.
  4. Dec 13, 2008 #3
    Thank you for your help.

    I'm trying to solve this without using groups or Lagrange's theorem (we didn't learn any of that in class).

    I made progress but I still am a little hung up at one (probably extremely trivial) detail.

    Let S be the set of units {x_1,...,x_n} and let a be a unit of the ring. T = {ax_1,...,ax_n} are all different and therefore T = S, which implies the products (call it z) are the same. a^n*z = z implies a^n = 1.

    I am not sure how to prove that all members of T = {ax_1,...,ax_n} are all unique, although I think it's pretty intuitive..
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Units of finite rings
  1. Units in a ring (Replies: 0)

  2. Units in a matrix ring (Replies: 4)