Results 1 to 2 of 2

Math Help - Checking answer of ring and ideal

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    83

    Checking answer of ring and ideal

    Let I be a subset of R, where R is a commutative ring and let I be an ideal of R.
    Show that:
    (i) I = R <=> 1 belongs to I <=> R/I = 0

    (ii) For a in R, (a) = 0 ={0} <=> a = 0
    and (a) = R <=> a in U(R)
    Where (a) = Ra = {ra: r in R} is the principal ideal generated by a

    This is my attempt:

    (i) (=>) Since I = R and 1 is in R, so 1 belongs to I
    (=>) 1 in R, then 1 + I = 0+I = R/I = 0
    Hence R/I = 0 = { 0+I}
    Coversely, (<=) R/I = 0, so 1+ I = 0 + I implies 1 belongs to I and since I in R then I = R
    Is that enough for this argument?

    Part(ii) I am not sure how to do it, please give some hints

    Thank you so much in advanced
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Since 0\in I and 0+I=I, from 1+I=0+I you can deduce 1=1+0\in 0+I=I, so it is ok.

    (a)=\{0\} \Leftrightarrow a=0 is simple, just use the definitions. For instance, (a)=\{0\} means for all r\in R,\ ra=0. Then with r=1\ ...

    (a)=R\Leftrightarrow a\in U(R) can be shown using (i): prove that 1\in (a)\Leftrightarrow a\in U(R) .


    In principal ideals, every element is a product of the generator with something in the ring: x belongs to (a) iff there is a b\in R such that ba=x.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ideal,nil,nilpotent ideal in prime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 24th 2011, 08:57 AM
  2. Ideal of the ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 27th 2011, 10:15 PM
  3. Ideal of ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 16th 2009, 10:58 PM
  4. ring/ideal
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 19th 2009, 03:52 PM
  5. Checking answer: is 0 a maximal ideal?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 31st 2009, 07:13 AM

Search Tags


/mathhelpforum @mathhelpforum