Primitive ideal
Encyclopedia
In mathematics
, a left primitive ideal in ring theory
is the annihilator
of a simple
left module
. A right primitive ideal is defined similarly. Note that (despite the name) 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a left primitive ideal in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
is the annihilator
Annihilator (ring theory)
In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
of a simple
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
left module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
. A right primitive ideal is defined similarly. Note that (despite the name) 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 ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.- Definition :...
.