Results 1 to 2 of 2

Thread: Commutative Ring, Subgroups, Subrings, Ideals

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    47

    Commutative Ring, Subgroups, Subrings, Ideals

    Let $\displaystyle \mathcal{F} $ be the set of functions $\displaystyle f:\mathbb{R} \ \rightarrow \ \mathbb{R}$. Given that $\displaystyle \mathcal{F} $ forms a commutative ring with indentity under (pointwise) addition and multiplication,

    (a) Find a subgroup of $\displaystyle \mathcal{F} $ which is not a subring. Why?
    (b) Find a subring of $\displaystyle \mathcal{F} $ which is not an ideal. Why?
    (c) Find an ideal of $\displaystyle \mathcal{F} $ which is not the zero ring or $\displaystyle \mathcal{F} $. Can this ideal be written as a principal idea? Why?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    i'll leave the "whys" for you to figure out!

    Quote Originally Posted by Jimmy_W View Post
    Let $\displaystyle \mathcal{F} $ be the set of functions $\displaystyle f:\mathbb{R} \ \rightarrow \ \mathbb{R}$. Given that $\displaystyle \mathcal{F} $ forms a commutative ring with indentity under (pointwise) addition and multiplication,

    (a) Find a subgroup of $\displaystyle \mathcal{F} $ which is not a subring. Why?
    set of all polynomials with coefficients in $\displaystyle \mathbb{R}$ and of degree at most 1. this is an additive subgroup which is not closed under multiplication.


    (b) Find a subring of $\displaystyle \mathcal{F} $ which is not an ideal. Why?
    set of all constant functions or, if you want the subring to have the identity element of $\displaystyle \mathcal{F},$ then $\displaystyle \mathbb{R}[x]$ would be an example.


    (c) Find an ideal of $\displaystyle \mathcal{F} $ which is not the zero ring or $\displaystyle \mathcal{F} $. Can this ideal be written as a principal idea? Why?
    fix an $\displaystyle a \in \mathbb{R}$ and define $\displaystyle I_a=\{f \in \mathcal{F}: \ f(a)=0 \}.$ this is a principal ideal. why?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Herstein: Subrings and Ideals
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Nov 14th 2011, 04:00 PM
  2. Subrings and subgroups of Z
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Sep 3rd 2011, 10:47 PM
  3. Ideals of ring and isomorphic ring :)
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 24th 2009, 03:23 AM
  4. Subrings and Ideals Questions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Apr 23rd 2009, 11:11 PM
  5. Prime Ideals of Commutative Rings
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 14th 2009, 07:27 PM

Search Tags


/mathhelpforum @mathhelpforum