# Maximum ideal

**Maximum ideal** is a term from algebra .

## definition

Let it be a ring . Then an ideal is called **maximal** if there is *a **maximal element* in the set of *all **real ideals* semi-ordered by the (set-theoretical) inclusion . That means for every real ideal :
* *

- From follows

In other words:

A true ideal is called maximal when there is no other true ideal of that contains entirely.

## Remarks

- The same applies to left and right ideals.
- With the help of Zorn's lemma one can show that every true ideal in a ring with one element 1 is contained in a maximal ideal.
- From this it follows in turn that every element of a commutative ring with 1 that is not a unit must be contained in a maximal ideal. In non-commutative rings this is i. A. wrong, as the example of the die rings over (oblique) bodies shows.
- Let be an ideal of the commutative ring with 1. The factor ring is a field if and only if is maximal. In particular, this means: The image of a ring homomorphism is a body if and only if its core is maximal.
- Rings can contain several maximal ideals. A ring that has only a single maximum left or right ideal is called a local ring . This is a two-sided ideal , and the quotient ring is called
**the**residue field of the ring called. - A maximal (two-sided) ideal of a ring is prime if and only if . In particular, is prime if contains a one element.

## Examples

- In the ring of whole numbers , every prime ideal except the zero ideal is maximal. However, this is generally not true; Integrity rings with this property are called (if they are not bodies) one-dimensional . All main ideal rings have this property.
- Let be the ring of continuous functions on the real numbers with pointwise multiplication. Consider the ring homomorphism

- In other words: the mapping that evaluates every function at position 0. The image of is , therefore, a body. Thus the core, i.e. the set of all functions with , is a maximum ideal.