semiprime ideal


Let R be a ring. An ideal I of R is a semiprime idealMathworldPlanetmath if it satisfies the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions:

(a) I can be expressed as an intersectionMathworldPlanetmath of prime idealsMathworldPlanetmathPlanetmath of R;

(b) if xR, and xRxI, then xI;

(c) if J is a two-sided idealMathworldPlanetmath of R and J2I, then JI as well;

(d) if J is a left idealMathworldPlanetmath of R and J2I, then JI as well;

(e) if J is a right ideal of R and J2I, then JI as well.

Here J2 is the product of ideals JJ.

The ring R itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.

A ring R is said to be a semiprime ring if its zero idealMathworldPlanetmathPlanetmath is a semiprime ideal.

Note that an ideal I of R is semiprime if and only if the quotient ringMathworldPlanetmath R/I is a semiprime ring.

Title semiprime ideal
Canonical name SemiprimeIdeal
Date of creation 2013-03-22 12:01:23
Last modified on 2013-03-22 12:01:23
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 11
Author antizeus (11)
Entry type Definition
Classification msc 16D25
Related topic NSystem
Defines semiprime ring
Defines semiprime