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.^[1] The set of all n × n matrices with entries in R is a matrix ring denoted M_n(R)^[2][3][4][5] (alternative notations: Mat_n(R)^[3] and R^n×n[6]). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs.

When R is a commutative ring, the matrix ring M_n(R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then the matrix rM is the matrix M with each of its entries multiplied by r.

• The set of all n × n
  square matrices
  over R, denoted M_n(R). This is sometimes called the "full ring of n-by-n matrices".
• The set of all upper
  triangular matrices
  over R.
• The set of all lower
  triangular matrices
  over R.
• The set of all
  isomorphic to the direct product
  of n copies of R.
• For any index set I, the ring of endomorphisms of the right R-module ${\textstyle M=\bigoplus _{i\in I}R}$ is isomorphic to the ring ${\displaystyle \mathbb {CFM} _{I}(R)}$^[citation needed] of column finite matrices whose entries are indexed by I × I and whose columns each contain only finitely many nonzero entries. The ring of endomorphisms of M considered as a left R-module is isomorphic to the ring ${\displaystyle \mathbb {RFM} _{I}(R)}$ of row finite matrices.
• If R is a
  absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring.^{[dubious – discuss]} Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.^{[dubious – discuss]} This idea can be used to represent operators on Hilbert spaces
  , for example.
• The intersection of the row-finite and column-finite matrix rings forms a ring ${\displaystyle \mathbb {RCFM} _{I}(R)}$.
• If R is
  matrix transposition
  .
• If A is a C*-algebra, then M_n(A) is another C*-algebra. If A is non-unital, then M_n(A) is also non-unital. By the Gelfand–Naimark theorem, there exists a Hilbert space H and an isometric *-isomorphism from A to a norm-closed subalgebra of the algebra B(H) of continuous operators; this identifies M_n(A) with a subalgebra of B(H^⊕n). For simplicity, if we further suppose that H is separable and A ${\displaystyle \subseteq }$ 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 ${\textstyle {\begin{pmatrix}pAp&pA(1-p)\\(1-p)Ap&(1-p)A(1-p)\end{pmatrix}}}$, 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 finite-dimensional simple associative algebras over the field C of
  biquaternions^[7] and modern authors would call tensors in C ⊗_R H, that was later shown to be isomorphic to M₂(C). One basis of M₂(C) consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by the identity matrix and the three Pauli matrices
  .
• 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).

• The matrix ring M_n(R) can be identified with the
  ring of endomorphisms of the free right R-module of rank n; that is, M_n(R) ≅ End_R(Rⁿ). Matrix multiplication
  corresponds to composition of endomorphisms.
• The ring M_n(D) over a
  semisimple ring
  . The rings ${\displaystyle \mathbb {CFM} _{I}(D)}$ and ${\displaystyle \mathbb {RFM} _{I}(D)}$ are not simple and not Artinian if the set I is infinite, but they are still
  full linear rings
  .
• The
  Artin–Wedderburn theorem states that every semisimple ring is isomorphic to a finite direct product
  ${\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})}$, for some nonnegative integer r, positive integers n_i, and division rings D_i.
• When we view M_n(C) as the ring of linear endomorphisms of Cⁿ, those matrices which vanish on a given subspace V form a
  left ideal. Conversely, for a given left ideal I of M_n(C) the intersection of null spaces
  of all matrices in I gives a subspace of Cⁿ. Under this construction, the left ideals of M_n(C) are in bijection with the subspaces of Cⁿ.
• There is a bijection 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 left R-modules and left M_n(R)-modules, and between the isomorphism classes of left ideals of R and left ideals of M_n(R). Identical statements hold for right modules and right ideals. Through Morita equivalence, M_n(R) inherits any Morita-invariant properties of R, such as being simple, Artinian, Noetherian, prime
  .

• If S is a subring of R, then M_n(S) is a subring of M_n(R). For example, M_n(Z) is a subring of M_n(Q).
• The matrix ring M_n(R) is commutative if and only if n = 0, R = 0, or R is commutative and n = 1. In fact, this is true also for the subring of upper triangular matrices. Here is an example showing two upper triangular 2 × 2 matrices that do not commute, assuming 1 ≠ 0 in R:

  ${\displaystyle {\begin{bmatrix}1&0\\0&0\end{bmatrix}}{\begin{bmatrix}1&1\\0&0\end{bmatrix}}={\begin{bmatrix}1&1\\0&0\end{bmatrix}}}$

  and

  ${\displaystyle {\begin{bmatrix}1&1\\0&0\end{bmatrix}}{\begin{bmatrix}1&0\\0&0\end{bmatrix}}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}.}$

• For n ≥ 2, the matrix ring M_n(R) over a
  nilpotent elements
  ; the same holds for the ring of upper triangular matrices. An example in 2 × 2 matrices would be

  ${\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\end{bmatrix}}={\begin{bmatrix}0&0\\0&0\end{bmatrix}}.}$

• The center of M_n(R) consists of the scalar multiples of the identity matrix, I_n, in which the scalar belongs to the center of R.
• The
  unit group
  of M_n(R), consisting of the invertible matrices under multiplication, is denoted GL_n(R).
• If F is a field, then for any two matrices A and B in M_n(F), the equality AB = I_n implies BA = I_n. 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).

In fact, R needs to be only a semiring for M_n(R) to be defined. In this case, M_n(R) is a semiring, called the matrix semiring. Similarly, if R is a commutative semiring, then M_n(R) is a matrix semialgebra.

For example, if R is the

• Central simple algebra
• Clifford algebra
• Hurwitz's theorem (normed division algebras)
• Generic matrix ring
• Sylvester's law of inertia