Question #203330

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.


1
Expert's answer
2021-06-17T12:40:02-0400

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


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


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