Primitive ideal
In mathematics, a left primitive ideal in ring theory is the annihilator of a simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
The quotient of a ring by a left primitive ideal is a left primitive ring.
Primitive spectrum
A primitive spectrum is a non-commutative analog[note 1] of a prime spectrum of a commutative ring.
Let A be a ring and the set of all primitive ideals of A. Then there is a topology on , 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 of A and thus there is a surjection
Example: a spectrum of a C*-algebra.
See also
Notes
- ↑ Although a primitive ideal is prime, a primitive ideal tends to be more of interest than a prime ideal in non-commutative ring theory.
References
- Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
- Isaacs, I. Martin (1994), Algebra, Brooks/Cole Publishing Company, ISBN 0-534-19002-2