Commutative Ring, Subgroups, Subrings, Ideals

**Jimmy_W** · May 6th 2009, 03:28 AM

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?

**NonCommAlg** · May 6th 2009, 03:56 AM

i'll leave the "whys" for you to figure out!

Originally Posted by Jimmy_W

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?