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Math Help - Ideal of the Ring of Continuous Functions

  1. #1
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    Ideal of the Ring of Continuous Functions

    Let T be the set of continous functions on the reals {f : R \rightarrow R}

    It's not hard to show that it forms a ring under "pointwise" addition and multiplication:

    (f+g)(x) = f(x) + g(x)
    (fg)(x) = f(x)g(x)

    Now, Let I = \{ f \in T : f(2) = 0\}

    I am trying to
    i) Show that I is an ideal in T.
    ii) Describe T/I using the 1st isomorphism theorem.

    Well, in i), I think that if we let f \in I, g \in T then (fg)(2) = f(2)g(2) = 0g(2) = 0 so fg \in I so it's an ideal? Also, it contains the 0 element and is closed under "subtraction".

    and in ii), I need to think of a ring homomorphism from T to some ring which has I as the kernel, but I can't seem to think what that would be!!

    Any help appreciated.... and if anyone could verify my reasoning in (i) that would be great!
    Thanks!!
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  2. #2
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    Quote Originally Posted by matt.qmar View Post
    Let T be the set of continous functions on the reals {f : R \rightarrow R}

    It's not hard to show that it forms a ring under "pointwise" addition and multiplication:

    (f+g)(x) = f(x) + g(x)
    (fg)(x) = f(x)g(x)

    Now, Let I = \{ f \in T : f(2) = 0\}

    I am trying to
    i) Show that I is an ideal in T.
    ii) Describe T/I using the 1st isomorphism theorem.

    Well, in i), I think that if we let f \in I, g \in T then (fg)(2) = f(2)g(2) = 0g(2) = 0 so fg \in I so it's an ideal? Also, it contains the 0 element and is closed under "subtraction".

    and in ii), I need to think of a ring homomorphism from T to some ring which has I as the kernel, but I can't seem to think what that would be!!

    Any help appreciated.... and if anyone could verify my reasoning in (i) that would be great!
    Thanks!!


    Look at the ring homomorphism \phi:T\rightarrow \mathbb{R}\,,\,\,\phi(f):=f(2)

    Tonio
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