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Maximal ideals are important because the Quotient Ring s of maximal ideals are Simple Ring s and in the special case of Unital Commutative Ring s even Field s. Rings which contain only one maximal ideal are called Local Ring s DEFINITION Given a ring ''R'' and a proper ideal ''I'' of ''R'' (that is ''I'' ≠ ''R''), ''I'' is called maximal ideal in ''R'' if there exists no other proper ideal ''J'' of ''R'' so that ''I'' ⊂ ''J''- An ideal S of a ring such that S≠ R is called a maximal ideal of R if there exists no proper ideals of R contaning S. EXAMPLES
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