Ideal (ring theory)

Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J . The factor ring of a maximal ideal is a field.
Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R , if ab is in I , then at least one of a and b is in I . The factor ring of a prime ideal is an integral domain.
Primary ideal: An ideal I is called primary ideal if for all a and b in R , if ab is in I , then at least one of a and b &lt;sup&gt; n &lt;/sup&gt; is in I for some natural number n . Every prime ideal is primary, but not conversely.
Principal ideal: An ideal generated by one element.
Primitive ideal: is the annihilator of a simple left module. A right primitive ideal is defined similarly. Note that (despite the name) left and right primitive ideals are always two-sided ideals. Factor rings constructed with primitive ideals are primitive rings.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".