High Quality Content by WIKIPEDIA articles! In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ? R, and whenever J is another ideal containing I as a subset, then either J = I or J = R. So there are no ideals "in between" I and R. Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. Rings which contain only one maximal ideal are called local rings.

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