Maximum ideal

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Maximum ideal is a term from algebra .


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.


  • 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.


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.