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    Ideal of continuous functions vanishing at a point

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    Let C[a,b] be the ring of complex valued continuous functions on closed and bounded interval [a,b]. You may assume that it is a ring under the operation of addition and multiplication of functions.

    Show that I_c = {f|f(c)= 0} is an ideal in this ring.

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    https://brainmass.com/math/complex-analysis/ideal-of-continuous-functions-vanishing-point-517336

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    Please see attachment for properly formatted copy.

    Question: I_c={f in C[a,b]|f(c)=0} is an ideal in the ring of complex valued continuous functions C[a,b].

    The ring C[a,b] is a commutative ring (product of functions is commutative). A set I in a ring R is called an ideal if
    - [(i)] (I,+) is a subgroup of (R,+).
    - ...

    Solution Summary

    It is proved that the set of continuous complex valued functions on an interval vanishing at a given point form an ideal in the ring of continuous functions.

    $2.19