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Maximal Ideals : Commutative Rings and Pointwise Operations ( Function Addition and Composition )

Let R be the set of all continuous functions from the set of real numbers into itself. Then R is a commutative ring with the following operations: (f+g)(x)=f(x) + g(x) and (fg)(x)=f(x)g(x) for all x.

Now let I be the set of all functions f(x) an element of R such that f(1)=0. Show that I is a maximal ideal of R.

Maximal ideal: I is a maximal ideal if for all ideals J of R such that I is contained in J is contained in R, then either J=I or J=R. In other words we cannot "squeeze" another ideal between I and the whole ring R.

Solution Preview

Proof:
First, I claim that I is an ideal.
For any f(x), g(x) in I, f(1)+g(1)=0+0=0 and f(1)g(1)=0*0=0. Thus I is a subring.
For any f(x) in I and g(x) in R, we have g(1)f(1)=0*g(1)=0. ...

Solution Summary

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