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Abstract Algebra: Ring Theory Problems

  1. Jun 4, 2009 #1
    Hello all,

    I am trying to work on some Ring Theory proofs and my Abstract Algebra is very minimal as I have not taken the class but need to look into it nonetheless. If anyone can figure these out for me I'd greatly appreciate it.

    Also, I am familiar in LaTeX typesetting but I don't know how to fluidly use it on this forum. If I enable the TeX can I use my typical math commands and implement conventional TeX'ing?

    1. The problem statement, all variables and given/known data

    This is the doozy for me:

    If R1,R2,...,Rn are commutative rings with identity, shows that U (R1 x R2 x ... x Rn) = U(R1) x U(R2) x ... x U(Rn)

    2. Relevant equations

    Identity: There exists some 1 in R such that a*1 = 1*a = a
    Commutativity: A, b in R such that ab = ba for all a, b.
    Unit: Must have identity, there exist some y in R such that xy = yx =1. Must be commutative.

    3. The attempt at a solution

    I've tried all sorts of debauchery but nothing seems to work. I have tried splitting this up and looking at it different ways, but nothing I've come up with is correct.

    1. The problem statement, all variables and given/known data

    Show that a finite commutative ring with no zero-divisors and at least two elements has identity.

    2. Relevant equations

    See above.

    3. The attempt at a solution

    So I need to show it is finite (easy), commutative, no zero-divisors, and and at least two elements has identity.

    I define some ring R with no zero-divisors and at least two elements. Let a,b be in R. There exists a*a=1; a*b= a (or b). If a*a=a, then a*a*b=a*b...

    Here I just keep running in circles.

    1. The problem statement, all variables and given/known data

    Prove that (x) in Q[x] is maximal.

    2. Relevant equations

    Q is the set of all rationals.
    http://en.wikipedia.org/wiki/Maximal_element" [Broken]

    3. The attempt at a solution

    Nothing. =[

    Thank you for all the help!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jun 5, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You use Latex on this board by enclosing it in [ tex ]...[ /tex ] or [ itex ]... [ /itex ] or [ latex] ... [ latex ] without the spaces.

    [tex]\frac{a}{b}[/tex]
    [itex]\frac{a}{b}[/itex]
    [itex]\frac{a}{b}[/itex]

    As for your problem, what do you mean by U (R1 x R2 x ... x Rn) = U(R1) x U(R2) x ... x U(Rn)? What is that "U"?
     
  4. Jun 5, 2009 #3
    For your second problem, think of the definition of a ring. It is a group, with two operations (R,+,.) ... What do you know about the operation of addition in the ring? i.e. what must (R,+) be?
     
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