primitive ideal (original) (raw)

Let R be a ring, and let I be an ideal of R. We say that I is a left (right) primitive idealif there exists a simple left (right) R-module Xsuch that I is the annihilator of X in R.

We say that R is a left (right) primitive ringif the zero ideal is a left (right) primitive ideal of R.

Note that I is a left (right) primitive ideal if and only if R/I is a left (right) primitive ring.