 Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals.Since a one sided maximal ideal "A" is not necessarily two-sided, the quotient "R"/"A" is not necessarily a ring, but it is a simple module over "R".