Question

Let F be the ring of all functions from R to R w.r.t. pointwise, addition and

multiplication. Let S be the set of all differentiable functions in F. Check whether S

is

i) a subring of F,

ii) an ideal of F.

Answer 1

i) Let us check whether $S$ is a subring of $F$. Let $f(x),g(x)\in S,$ that is $f(x),g(x)$ are differentiable functions. Then the function $f(x)-g(x)$ is also differentiable and $(f(x)-g(x))'=f'(x)-g'(x),$ and hence $f(x)-g(x)\in S.$ Also the function $f(x)\cdot g(x)$ is differentiable and $(f(x)\cdot g(x))'=f(x)g'(x)+f'(x)g(x),$ and hence $f(x)\cdot g(x)\in S.$ Therefore, we conclude that $S$ is a subring of $F.$

ii) Let us check whether $S$ is an ideal of $F$. Let $f(x)\notin S,$ for example $f(x)=|x|.$ Taking into account that the constant function $g:\R\to\R,\ g(x)=1,$ is differentiable, that is $g(x)\in S,$ but $g(x)\cdot f(x)=f(x)\notin S,$ we conclude that $S$ is not an ideal of $F$.