Matrix ring

# Matrix ring

Discussion

Encyclopedia
In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a matrix ring is any collection of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

forming 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...

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....

and matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings. Any subrings of these matrix rings are also called matrix rings.

In the case when R is a commutative ring, then the matrix ring Mn(R) is an associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

which may be called a matrix algebra. In this situation, if M is a matrix and r is in R, then the matrix Mr is the matrix M with each of its entries multiplied by R.

It is assumed throughout that R is an associative ring with a unit 1 ≠ 0, although matrix rings can be formed over rings without unity.

## Examples

• The set of all n×n matrices over an arbitrary ring R, denoted Mn(R). This is usually referred to as the "full ring of n by n matrices". These matrices represent endomorphisms of the free module Rn.

• The set of all upper (or set of all lower) triangular matrices over a ring.

• If R is any ring with unity, then the ring of endomorphisms of as a right R module is isomorphic to the ring of column finite matrices whose entries are indexed by , and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices whose rows each only have finitely many nonzero entries.

• If R is a normed ring, then the condition of row or column finiteness in the previous point can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. This idea can be used to represent operators on Hilbert spaces, for example.

• The intersection of the row and column finite matrix rings also forms a ring, which can be denoted by .

• The algebra M2(R) of 2 × 2 real matrices is a simple example of a non-commutative associative algebra. Like the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s, it has dimension 4 over R, but unlike the quaternions, it has zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...

s, as can be seen from the following product of the matrix unit
Matrix unit
In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. The topic is comparatively obscure within linear algebra, because it entirely ignores the numeric properties of matrices; it is mostly encountered in the...

s: E11E21 = 0, hence it is not a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

. Its invertible elements are nonsingular matrices and they form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

GL(2,R).

• If R is commutative
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....

, the matrix ring has a structure of a *-algebra over R, where the involution * on Mn(R) is the matrix transposition.

• Complex matrix algebras Mn(C) are, up to isomorphism, the only simple associative algebras over the field C of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s. For n = 2, the matrix algebra M2(C) plays an important role in the theory of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

. It has an alternative basis given by the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

and the three Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

. M2(C) was the scene of early abstract algebra in the form of biquaternions.

• A matrix ring over a field is a Frobenius algebra
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in...

, with Frobenius form given by the trace of the product: σ(A,B)=tr(AB).

## Structure

• The matrix ring Mn(R) can be identified with the ring of endomorphisms of the free R-module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...

of rank n, Mn(R) ≅ EndR(Rn). The procedure for matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

can be traced back to compositions of endomorphisms in this endomorphism ring.

• The ring Mn(D) over a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

D is 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...

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...

, a special type of semisimple ring. The rings and are not simple and not Artinian if the set is infinite, however they are still full linear rings.

• In general, every semisimple ring is isomorphic to a finite direct product of full matrix rings over division rings, which may have differing division rings and differing sizes. This classification is given by the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...

.

• There is a one-to-one correspondence between the two-sided ideals
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"....

of Mn(R) and the two-sided ideals of R. Namely, for each ideal I of R, the set of all n×n matrices with entries in I is an ideal of Mn(R), and each ideal of Mn(R) arises in this way. This implies that Mn(R) is 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...

if and only if R is simple. For n ≥ 2, not every left ideal or right ideal of Mn(R) arises by the previous construction from a left ideal or a right ideal in R. For example, the set of matrices whose columns with indices 2 through n are all zero forms a left ideal in Mn(R).

• The previous ideal correspondence actually arises from the fact that the rings R and Mn(R) are Morita equivalent. Roughly speaking, this means that the category of left R modules and the category of left Mn(R) modules are very similar. Because of this, there is a natural bijective correspondence between the isomorphism classes of the left R-modules and the left Mn(R)-modules, and between the isomorphism classes of the left ideals of R and Mn(R). Identical statements hold for right modules and right ideals. Through Morita equivalence, Mn(R) can inherit any properties of R which are Morita invariant, such as being 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...

, 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...

, Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

, prime
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...

and numerous other properties as given in the Morita equivalence
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 :...

article.

## Properties

• The matrix ring Mn(R) is commutative
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....

if and only if n = 1 and R is commutative
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....

. As an example for 2×2 matrices which do not commute,
and
. This example is easily generalized to n×n matrices.
• For n ≥ 2, the matrix ring Mn(R) has zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...

s. An example in 2×2 matrices would be.
• The center
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...

of a matrix ring over a ring R consists of the matrices which are scalar multiples of the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

, where the scalar belongs to the center
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...

of R.

• In linear algebra, it is noted that over a field F, Mn(F) has the property that for any two matrices A and B, AB=1 implies BA=1. This is not true for every ring R though. A ring R whose matrix rings all have the mentioned property is known as a stably finite ring or sometimes weakly finite ring .