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Math Help - Algebra, Problems For Fun (14)

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    Algebra, Problems For Fun (14)

    Throughout R is a ring with identity element.

    Definition 1: R is called left (right) artinian if any descending chain of the left (right) ideals of R stabilizes. Clearly every finite ring is left and right artinian.

    Definition 2: R is called reduced if it has no non-zero nilpotent element, i.e. if x^n=0, for some x \in R and integer n \geq 2, then x=0.

    Artin-Wedderburn Theorem: Every reduced left (or right) artinian ring R is a finite direct product of matrix rings over some division rings, that is:

    R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k),

    for some integers n_j \geq 1 and division rings D_j.

    Wedderburn's little theorem: Every finite division ring is a field.


    Question: Give a short proof of the following problem, which is a special case of Jacobson's difficult theorem:

    Let R be a finite ring and suppose that for any x \in R, there exists an integer n(x) \geq 2 such that x^{n(x)}=x. Prove that R is commutative.
    Last edited by NonCommAlg; June 5th 2009 at 04:01 PM.
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