Semiprimitive ring
Encyclopedia
In mathematics
, especially in the area of algebra
known as ring theory
, a semiprimitive ring is a type of ring more general than a semisimple ring, but where simple module
s still provide enough information about the ring. Important rings such as the ring of integers are semiprimitive, and an artinian
semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive ring
s, which are described by the Jacobson density theorem
. The quotient of every ring by its Jacobson radical
is semiprimitive, allowing every ring to be understood to some extent through semiprimitive rings.
is the zero ideal.
A ring is semiprimitive if and only if it has a faithful semisimple left module
. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module.
A ring is semiprimitive if and only if it is a subdirect product of left primitive ring
s.
A commutative ring
is semiprimitive if and only if it is a subdirect product of fields
, .
A left artinian ring
is semiprimitive if and only if it is semisimple, .
Jacobson
himself has defined a ring to be "semisimple" if and only if it is a subdirect product
of simple ring
s, . However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, .
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...
, especially in the area of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
known as 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...
, a semiprimitive ring is a type of ring more general than a semisimple ring, but where simple module
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...
s still provide enough information about the ring. Important rings such as the ring of integers are semiprimitive, and an artinian
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of 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 :...
s, which are described by the Jacobson density theorem
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R....
. The quotient of every ring by its Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
is semiprimitive, allowing every ring to be understood to some extent through semiprimitive rings.
Definition
A ring is called semiprimitive or Jacobson semisimple if its Jacobson radicalJacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
is the zero ideal.
A ring is semiprimitive if and only if it has a faithful semisimple left module
Semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring which is a semisimple module over itself is known as an artinian semisimple ring...
. The semiprimitive property is left-right symmetric, and so a ring is semiprimitive if and only if it has a faithful semisimple right module.
A ring is semiprimitive if and only if it is a subdirect product of 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 :...
s.
A commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
is semiprimitive if and only if it is a subdirect product of fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, .
A left artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
is semiprimitive if and only if it is semisimple, .
Examples
- The ring of integers is semiprimitive, but not semisimple.
- Every primitive ring is semiprimitive.
- The product of two fields is semiprimitive but not primitive.
- Every von Neumann regular ringVon Neumann regular ringIn mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...
is semiprimitive.
Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
himself has defined a ring to be "semisimple" if and only if it is a subdirect product
Subdirect product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product...
of simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
s, . However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, .