Primitive ring
Encyclopedia
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 ring
s of vector spaces and Weyl algebras over fields of characteristic zero.
R is said to be a left primitive ring if and only if it has a faithful simple
left R-module
. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in . Another example found by Jategaonkar showing the distinction can be found in
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero twosided ideal
. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined by the Jacobson density theorem
: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space
over a division ring.
Another equivalent definition states that a ring is left primitive if and only if it is a prime ring
with a faithful left module of finite length
.
s and prime ring
s. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring
, it is known that the conditions "left primitive", "right primitive", "prime", and "simple
" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive"="right primitive"="prime".
A commutative ring
is left primitive if and only if it is a field
.
Being left primitive is a Morita invariant property
.
R with unity is both left and right primitive. (However, a simple non-unital ring, may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module
R/M is a simple left R-module, and that its annihilator
is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields with characteristic
zero are primitive, and since they are domain
s, they are examples without minimal one-sided ideals.
where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socle
soc(RR)≠{0}. Through linear algebra arguments, it can be shown that
is isomorphic to the ring of row finite matrices 
, where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D.
Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
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 left primitive ring is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
which has a faithful simple left module. Well known examples include endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
s of vector spaces and Weyl algebras over fields of characteristic zero.
Definition
A ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R is said to be a left primitive ring if and only if it has a faithful 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 R-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 ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in . Another example found by Jategaonkar showing the distinction can be found in
An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero twosided ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined 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....
: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a division ring.
Another equivalent definition states that a ring is left primitive if and only if it is a prime ring
Prime ring
In abstract algebra, a non-trivial ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. Prime ring can also refer to the subring of a field determined by its characteristic...
with a faithful left module of finite length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
.
Properties
One sided primitive rings are both semiprimitive ringSemiprimitive ring
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 modules still provide enough information about the ring. Important rings such as the ring of integers are semiprimitive, and an...
s and prime ring
Prime ring
In abstract algebra, a non-trivial ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. Prime ring can also refer to the subring of a field determined by its characteristic...
s. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For 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...
, it is known that the conditions "left primitive", "right primitive", "prime", and "simple
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...
" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive"="right primitive"="prime".
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 left primitive if and only if it is a field
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...
.
Being left primitive is a Morita invariant property
Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.- Motivation :...
.
Examples
Every simple ringSimple 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...
R with unity is both left and right primitive. (However, a simple non-unital ring, may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module
Quotient module
In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...
R/M is a simple left R-module, and that its 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...
is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields with characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
zero are primitive, and since they are domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
s, they are examples without minimal one-sided ideals.
Full linear rings
A special case of primitive rings is that of full linear rings. A left full linear ring is the ring of all linear transformations of an infinite dimensional left vector space over a division ring. (A right full linear ring differs by using a right vector space instead.) In symbols, where V is a vector space over a division ring D. It is known that R is a left full linear ring if and only if R is von Neumann regular, left self-injective with socleSocle (mathematics)
-Socle of a group:In the context of group theory, the socle of a group G, denoted Soc, is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups...
soc(RR)≠{0}. Through linear algebra arguments, it can be shown that
is isomorphic to the ring of row finite matrices , where I is an index set whose size is the dimension of V over D. Likewise right full linear rings can be realized as column finite matrices over D.Using this we can see that there are non-simple left primitive rings. By the Jacobson Density characterization, a left full linear ring R is always left primitive. When dimDV is finite R is a square matrix ring over D, but when dimDV is infinite, the set of finite rank linear transformations is a proper two-sided ideal of R, and hence R is not simple.