Results 1 to 3 of 3

Math Help - properties of radical of an ideal

  1. #1
    Junior Member
    Joined
    Feb 2011
    Posts
    72

    properties of radical of an ideal

    Hello everybody;

    Let R be commutative ring with unity and let I and J be two ideals of R Show that

    \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by student2011 View Post
    Hello everybody;

    Let R be commutative ring with unity and let I and J be two ideals of R Show that

    \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}
    \sqrt{I+J} \subseteq \sqrt{\sqrt{I}+\sqrt{J}} is obvious. for the other inclusion, let a \in  \sqrt{\sqrt{I}+\sqrt{J}}. then there exists a positive integer n such that a^n = b+ c, for some b \in \sqrt{I} and c \in \sqrt{J}. so b^r \in I and c^s \in J, for some positive integers r, s. now show that a^{n(r+s)} \in I + J and thus a \in \sqrt{I+J}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2011
    Posts
    72
    Thank you very much, I catch the point. In order to show that a^{n(r+s)}\in I+J, we do the following:

    (b+c)^{r+s}=\sum_{k=0}^{r+s}\binom{r+s}{k}\times b^{r+s-k}\times c^{k} \in I+J


    This shows that a^{n(r+s)} \in I+J
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. properties of radical ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 12th 2011, 12:46 PM
  2. Some Radical Ideal proofs
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 21st 2011, 08:46 PM
  3. Radical ideal
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 14th 2009, 10:07 PM
  4. Radical of an ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 7th 2008, 07:51 PM
  5. Radical and Ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 6th 2008, 07:26 PM

Search Tags


/mathhelpforum @mathhelpforum