Let F be the ring of all functions mapping R into R and having derivatives of all orders. Differentiation gives a map ∅ : F → F where δ(f(x)) = f'(x). Is δ a homomorphism? Why? Give the connection between this exercise and Example 26.12.

Data from Example 26.12

Let F be the ring of all functions mapping R. into R, and let C be the subring of F consisting of all the constant functions in F. Is C an ideal in F? Why?

Solution: It is not true that the product of a constant function with every function is again a constant function. For example, the product of sin x and 2 is the function 2 sin x. Thus C is not an ideal of F.