In the branch of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
known as
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a left primitive ring is a
ring which has a
faithful simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
left
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
. Well known examples include
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
s of
vector spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and
Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form
: f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X).
More prec ...
s over
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
of
characteristic zero.
Definition
A ring ''R'' is said to be a left primitive ring if it has a
faithful simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
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 two-sided
ideals. The analogous definition for right primitive rings is also valid.
The structure of left primitive rings is completely determined by the
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 .
The theorem can be applied to show that any primitive ring can be v ...
: A ring is left primitive if and only if it is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of the
ring of endomorphisms
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of a
left vector space over a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
.
Another equivalent definition states that a ring is left primitive if and only if it is a
prime ring
In abstract algebra, a nonzero 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. This definition can be regarded as a simultaneous generaliz ...
with a faithful left module of
finite length (
Ex. 11.19, p. 191.
Properties
One-sided primitive rings are both
semiprimitive ring
In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about ...
s and
prime ring
In abstract algebra, a nonzero 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. This definition can be regarded as a simultaneous generaliz ...
s. Since the
product ring
In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in ...
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 In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
, it is known that the conditions "left primitive", "right primitive", "prime", and "
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
" are all equivalent, and in this case it is a
semisimple ring
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 that is a semisimple module over itsel ...
isomorphic to a square
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive" = "right primitive" = "prime".
A
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is left primitive if and only if it is a
field.
Being left primitive is a
Morita invariant property.
Examples
Every
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
''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
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
''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 and therefore ''R''/''M'' is a faithful left ''R''-module.
Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form
: f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X).
More prec ...
s over fields of
characteristic zero are primitive, and since they are
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
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 transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s 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
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...
,
left self-injective with
socle soc(
''R''''R'') ≠ . Through
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
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 dim
''D''''V'' is finite ''R'' is a square matrix ring over ''D'', but when dim
''D''''V'' is infinite, the set of finite rank linear transformations is a proper two-sided ideal of ''R'', and hence ''R'' is not simple.
See also
*
primitive ideal
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.
Primitive ideals ar ...
References
p. 1000 errata*
*
*{{citation
, last=Rowen
, first=Louis H.
, title=Ring theory. Vol. I
, series=Pure and Applied Mathematics
, volume=127
, publisher=Academic Press Inc.
, place=Boston, MA
, year=1988
, pages=xxiv+538
, isbn=0-12-599841-4
, mr=940245
Ring theory
Algebraic structures