Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By Deveno

Math Help - Polynomial rings - zero divisors in R and r[x]

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    Polynomial rings - zero divisors in R and r[x]

    Show that if R has no zero divisors then R[x] also has no zero divisors.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    Re: Polynomial rings - zero divisors in R and r[x]

    Thinking about this problem ...

    Let  f(x) = a_n x^n + ... + a_1 x + a_0

    and  g(x) - b_n x^n + ... + b_1 x + b_0

    Then f(x)g(x) = c_{m+n} x^{m+n) + c_{m+n- 1} x^{m+n-1) + ... c_1 x + c_0

    where c_k = a_kb_0 + a_{k-1}b_1 + ... + a_1b_{k-1} + a_0b_k

    But c_k cannot equal zero since all of the a_i b_j \ne 0 since R has no zero divisors

    Thus R[x] has no zero divisors

    Can someone please confirm that this solution is OK?

    Peter
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: Polynomial rings - zero divisors in R and r[x]

    well, no.

    ck is a sum of elements, so there is no reason to suppose it can't be 0, even if all the terms in the sum are non-zero.

    however, you're close...

    cm+n = ambn ≠ 0, since am ≠ 0, and bn ≠ 0.

    since this is the leading coefficient of f(x)g(x), f(x)g(x) cannot be the 0-polynomial.
    Thanks from Bernhard
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Polynomial rings - zero divisors
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 27th 2012, 02:19 PM
  2. Rings and zero divisors
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 28th 2011, 08:58 AM
  3. Replies: 1
    Last Post: October 7th 2010, 07:38 PM
  4. Rings with unity and zero-divisors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 8th 2009, 07:45 AM
  5. Rings, and zero divisors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 2nd 2009, 10:27 PM

Search Tags


/mathhelpforum @mathhelpforum