# Commutative Ring, Subgroups, Subrings, Ideals

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• May 6th 2009, 03:28 AM
Jimmy_W
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?
• May 6th 2009, 03:56 AM
NonCommAlg
i'll leave the "whys" for you to figure out!

Quote:

Originally Posted by Jimmy_W
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.

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(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.

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(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?