Page "Maximal ideal" Paragraph 21

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

The result is also true if " ideal " is replaced with " right ideal " or " left ideal ".

More generally, it is true that every nonzero finitely generated module has a maximal submodule.

Suppose I is an ideal which is not R ( respectively, A is a right ideal which is not R ).

Then R / I is a ring with unity, ( respectively, R / A is a finitely generated module ), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal ( respectively maximal right ideal ) of R containing I ( respectively, A ).