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Math Help - Commutative Ring, Subgroups, Subrings, Ideals

  1. #1
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    Commutative Ring, Subgroups, Subrings, Ideals

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

    (a) Find a subgroup of  \mathcal{F} which is not a subring. Why?
    (b) Find a subring of  \mathcal{F} which is not an ideal. Why?
    (c) Find an ideal of  \mathcal{F} which is not the zero ring or  \mathcal{F} . Can this ideal be written as a principal idea? Why?
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  2. #2
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    i'll leave the "whys" for you to figure out!

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

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


    (b) Find a subring of  \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 \mathcal{F}, then \mathbb{R}[x] would be an example.


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