It turns out that the maximal ideal respects the order on "'Q"'.

By the above, any maximal ideal is prime.

In commutative rings, there is no difference, and one speaks simply of maximal ideals.

Rings which contain only one maximal ideal are called local rings.

Maximal ideals are in a sense easier to look for than annihilators of modules.

Every radical ideal is an intersection of maximal ideals.

The filtration of a local ring by the powers of its maximal ideal.

Any quotient of a ring by a maximal ideal is a simple ring.

The maximal ideal of A is principal.

Frequently, R is a local ring and m is then its unique maximal ideal.

Every maximal ideal is closed for the -adic topology.

Thus the closed points of Spec(R) are precisely the maximal ideals.

An integral domain is equal to the intersection of its localizations at maximal ideals.

In all unital rings, maximal ideals are prime.

In a commutative ring with unity, every maximal ideal is a prime ideal.

Every unital ring other than the trivial ring contains a maximal ideal.

Maximal ideals of a Boolean algebra are the same as prime ideals.

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

For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.

Every countable commutative ring has a maximal ideal.

Then R contains a maximal ideal.

In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal.

Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal .

In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number.

A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.