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Thursday, June 14, 2007

Maximal ideal

Let R be a ring with identity. A proper ideal I ⊆ R is a maximal ideal if I is not a proper subset of any other ideal of R.

All maximal ideals are prime ideals. If R is commutative, an ideal I ⊂ R is maximal if and only if the quotient ring R is a field.

Let m, n be ideals of Z.
I = (6) = {..., -12, -6, 0, 6, 12, 18, 24, ...}
J = (3) = {..., -6, -3, 0, 3, 6, 9, 12, 15, ...}
The ideal m is not maximal because I ⊂ J, while J is maximal because there is no ideal K such that J ⊂ K.

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