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Math Help - Finitely Generated k-algebra - Nature of the finite generation - basic question

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    Super Member Bernhard's Avatar
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    Finitely Generated k-algebra - Nature of the finite generation - basic question

    In Dummit and Foote Chapter 15 on page 657 we find the following definition of a k-algebra:

    Let k be a field. A ring R is a k-algebra if k is contained in the centre of R and the identity of k is the identity of R.

    This defintion is followed by the definition of a finitely generated k-algebra:

    The ring R is a finitely generated k-algebra if R is generated as a ring by k together with some finite set  r_1, r_2, ... ... , r_n of elements of R.

    I wish to make sure that I understand the idea of the finite generation of the ring R.

    I am assuming that we have as generators, the set of elements of k, namely  k_1, k_2, k_3, ... ... (may be infinite!)

    and also the ring elements  r_1, r_2, ... ... , r_n

    So the generated elements would be the those elements themselves plus the elements formed by combining these elements using (possibly repeated) the ring operations of addition and multiplication ...

    So typical generated elements would include elements like

     r_1 + r_2,  r_1 + k_1,   r_1k_1, r_1k_2 + r_2k_3, r_1r_2 + k_1k_4 + r_3k_4, r_1r_2k_1, r_2r_3k_2 + k_1k_2r_3, ... ... and so on ...

    Does this look right? Can someone please confirm that I am understanding this correctly.

    Peter
    Last edited by Bernhard; August 30th 2013 at 02:56 AM.
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