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A commutative ring is called a local ring if it has a unique maximal ideal

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A commutative ring 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 commutative ring with one in which the set of nonunits forms an ideal M, then R is a local ring with unique maximal ideal M.