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Thread: maximal ideal in commutative algebra

  1. #1
    Member Mauritzvdworm's Avatar
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    maximal ideal in commutative algebra

    I would like to show that every commutative algebra contains proper maximal ideals

    My plan is to create en increasing sequence of proper ideals in the commutative algebra A of the sort I_1\subseteq I_2\subseteq\dots such that I=\bigcup^{\infty}_{n=1}I_{n}
    then to show that every other porper ideal J of A will be contained in the union of some finite collection of these proper ideals I_{n} then I will be a maximal ideal.

    My problem is that I didn't really make any use of the commutative structure of the algebra, any ideas?
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  2. #2
    Senior Member
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    Quote Originally Posted by Mauritzvdworm View Post
    I would like to show that every commutative algebra contains proper maximal ideals

    My plan is to create en increasing sequence of proper ideals in the commutative algebra A of the sort I_1\subseteq I_2\subseteq\dots such that I=\bigcup^{\infty}_{n=1}I_{n}
    then to show that every other porper ideal J of A will be contained in the union of some finite collection of these proper ideals I_{n} then I will be a maximal ideal.

    My problem is that I didn't really make any use of the commutative structure of the algebra, any ideas?
    See link1 and link2.
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  3. #3
    MHF Contributor
    Opalg's Avatar
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    Notice also that when you are taking the union of an increasing chain of proper ideals, you need to ensure that the union is still a proper ideal. If the algebra is unital then this is not a problem, since none of the ideal contains the identity element and so neither does their union. But in the case of a nonunital algebra you need to use the concept of a modular ideal.

    It does not matter that you didn't make any use of the commutative structure of the algebra, because the result is equally valid in the case of a noncommutative algebra (though in that case you need to specify whether you are using left, right, or two-sided, ideals).
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