Matrix ring

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of n × n matrices with entries from R is a matrix ring denoted M_n(R), as well as some subsets of infinite matrices which form infinite matrix rings. Any subring of a matrix ring is a matrix ring.

When R is a commutative ring, the matrix ring M_n(R) is an associative algebra, and may be called a matrix algebra. For this case, 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.

This article assumes 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 M_n(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 $M=\bigoplus _{i\in I}R$ as a right R-module is isomorphic to the ring of column finite matrices $\mathbb{CFM}_I(R)\,$ whose entries are indexed by I × I, 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 $\mathbb {RFM} _{I}(R)$ whose rows each only have finitely many nonzero entries.
If R is a Banach algebra, 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 ${\mathbb {RCFM}}_{I}(R)$ .
The algebra M₂(R) of 2 × 2 real matrices, which is isomorphic to the split-quaternions, is a simple example of a non-commutative associative algebra. Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: E₁₁E₂₁ = 0, hence it is not a division ring. Its invertible elements are nonsingular matrices and they form a group, the general linear group GL(2, R).
If R is commutative, the matrix ring has a structure of a *-algebra over R, where the involution * on M_n(R) is the matrix transposition.
If A is a C*-algebra, then M_n(A) consists of n-by-n matrices with entries from the C*-algebra A, which is itself a C*-algebra. If A is non-unital, then M_n(A) is also non-unital. Viewing A as a norm-closed subalgebra of the continuous operators B(H) for some Hilbert space H (that there exists such a Hilbert space and isometric *-isomorphism is the content of the Gelfand-Naimark theorem), we can identify M_n(A) with a subalgebra of B(H^{$\oplus n$}). For simplicity, if we further suppose that H is separable and A $\subseteq$ B(H) is a unital C*-algebra, we can break up A into a matrix ring over a smaller C*-algebra. One can do so by fixing a projection p and hence its orthogonal projection 1 - p; one can identify A with ${\begin{pmatrix}pAp&pA(1-p)\\(1-p)Ap&(1-p)A(1-p)\end{pmatrix}}$ , where matrix multiplication works as intended because of the orthogonality of the projections. In order to identify A with a matrix ring over a C*-algebra, we require that p and 1 - p have the same ″rank″; more precisely, we need that p and 1 - p are Murray-von Neumann equivalent, i.e. there exists a partial isometry u such that p = uu* and 1 - p = u*u. One can easily generalize this to matrices of larger sizes.
Complex matrix algebras M_n(C) are, up to isomorphism, the only simple associative algebras over the field C of complex numbers. For n = 2, the matrix algebra M₂(C) plays an important role in the theory of angular momentum. It has an alternative basis given by the identity matrix and the three Pauli matrices. M₂(C) was the scene of early abstract algebra in the form of biquaternions.
A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: σ(A, B) = tr(AB).

Structure

The matrix ring M_n(R) can be identified with the ring of endomorphisms of the free R-module of rank n, M_n(R) ≅ End_R(Rⁿ). The procedure for matrix multiplication can be traced back to compositions of endomorphisms in this endomorphism ring.
The ring M_n(D) over a division ring D is an Artinian simple ring, a special type of semisimple ring. The rings $\mathbb{CFM}_I(D)$ and $\mathbb{RFM}_I(D)$ are not simple and not Artinian if the set I 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.
There is a one-to-one correspondence between the two-sided ideals of M_n(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 M_n(R), and each ideal of M_n(R) arises in this way. This implies that M_n(R) is simple if and only if R is simple. For n ≥ 2, not every left ideal or right ideal of M_n(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 M_n(R).
The previous ideal correspondence actually arises from the fact that the rings R and M_n(R) are Morita equivalent. Roughly speaking, this means that the category of left R modules and the category of left M_n(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 M_n(R)-modules, and between the isomorphism classes of the left ideals of R and M_n(R). Identical statements hold for right modules and right ideals. Through Morita equivalence, M_n(R) can inherit any properties of R which are Morita invariant, such as being simple, Artinian, Noetherian, prime and numerous other properties as given in the Morita equivalence article.

Properties

The matrix ring M_n(R) is commutative if and only if n = 1 and R is commutative. In fact, this is also true for the subring of upper triangular matrices. Here is an example for 2×2 matrices (in fact, upper triangular matrices) which do not commute:

\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}\,

and $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\,$ . This example is easily generalized to n×n matrices.

For n ≥ 2, the matrix ring M_n(R) has zero divisors and nilpotent elements, and again, the same thing can be said for the upper triangular matrices. An example in 2×2 matrices would be

\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\,

The center of a matrix ring over a ring R consists of the matrices which are scalar multiples of the identity matrix, where the scalar belongs to the center of R.
In linear algebra, it is noted that over a field F, M_n(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 (Lam 1999, p. 5).

Diagonal subring

Let D be the set of diagonal matrices in the matrix ring M_n(R), that is the set of the matrices such that every nonzero entry, if any, is on the main diagonal. Then D is closed under matrix addition and matrix multiplication, and contains the identity matrix, so it is a subalgebra of M_n(R).

As an algebra over R, D is isomorphic to the direct product of n copies of R. It is a free R-module of dimension n. The idempotent elements of D are the diagonal matrices such that the diagonal entries are themselves idempotent.

Two dimensional diagonal subrings

When R is the field of real numbers, then the diagonal subring of M₂(R), is isomorphic to split-complex numbers. When R is the field of complex numbers, then the diagonal subring is isomorphic to bicomplex numbers. When R = ℍ, the division ring of quaternions, then the diagonal subring is isomorphic to the ring of split-biquaternions, presented in 1873 by William K. Clifford.

Matrix semiring

In fact, R only needs to be a semiring for M_n(R) to be defined. In this case, M_n(R) is a semiring. If R={0,1} with 1+1=1, then M_n(R) is the semiring of binary relations on an n-element set with union as addition and composition as multiplication.

References

Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5

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