Prime ideal

This article is about ideals in ring theory. For prime ideals in order theory, see Ideal (order theory) § Prime ideals.
Not to be confused with Primary ideal.
A Hasse diagram of a portion of the lattice of ideals of the integers The purple nodes indicate prime ideals. The purple and green nodes are semiprime ideals, and the purple and blue nodes are primary ideals.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.[1][2] The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

Primitive ideals are prime, and prime ideals are both primary and semiprime.

  1. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
  2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.