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Math Help - noetherian ring help

  1. #1
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    noetherian ring help

    Could you please help or even just give me some advice. I have to prove/disprove the claim that every finite ring is noetherian.
    I know that a ring R is noetherian, if every ideal of R is finitely generated. Also I know there are many infinite rings which are noetherian such as the integers or afield like the complex numbers.

    Thanks if anyone can help me in proving this.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: noetherian ring help

    If an ideal of R is not finitely generated, does that not imply an infinite number of elements in the ring R?, since what does it mean for a Ideal to be generated by elements?
    There exists a set of a_1,...,a_n \in I such that I = \{a_1R + ... + a_nR\} . if a_1,...,a_k \in I forms an infinite set, since  I \subset R, a_1,...a_k \in R , so R must be infinite.
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