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?