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    Local Rings and Maximal Ideals

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    A commutative ring R is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal M then every element of (R-M) is a unit.

    Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.

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    Solution Summary

    Local rings and maximal ideals are investigated.