In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module.
For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
The primitive spectrum of a ring is a non-commutative analog[note 1] of the prime spectrum of a commutative ring.
, called the Jacobson topology, defined so that the closure of a subset T is the set of primitive ideals of A containing the intersection of elements of T. Now, suppose A is an associative algebra over a field.
Then, by definition, a primitive ideal is the kernel of an irreducible representation