Inabstract algebra, amatrix ring is a set ofmatrices with entries in aringR that form a ring undermatrix addition andmatrix multiplication.^[1] The set of alln ×n matrices with entries inR is a matrix ring denoted M_n(R)^[2][3][4][5] (alternative notations: Mat_n(R)^[3] andR^n×n[6]). Some sets of infinite matrices forminfinite matrix rings. A subring of a matrix ring is again a matrix ring. Over arng, one can form matrix rngs.

WhenR is a commutative ring, the matrix ring M_n(R) is anassociative algebra overR, and may be called amatrix algebra. In this setting, ifM is a matrix andr is inR, then the matrixrM is the matrixM with each of its entries multiplied byr.

• The set of alln ×nsquare matrices overR, denoted M_n(R). This is sometimes called the "full ring ofn-by-n matrices".
• The set of all uppertriangular matrices overR.
• The set of all lowertriangular matrices overR.
• The set of alldiagonal matrices overR. Thissubalgebra of M_n(R) isisomorphic to thedirect product ofn copies ofR.
• For any index setI, the ring of endomorphisms of the rightR-module$M=\underset{i\in I}{⨁}R$ is isomorphic to the ring${\displaystyle {\mathbb{C}\mathbb{F}\mathbb{M}}_I(R)}$^{[citation needed]} ofcolumn finite matrices whose entries are indexed byI ×I and whose columns each contain only finitely many nonzero entries. The ring of endomorphisms ofM considered as a leftR-module is isomorphic to the ring${\displaystyle {\mathbb{R}\mathbb{F}\mathbb{M}}_I(R)}$ ofrow finite matrices.
• IfR is aBanach 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.^{[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 representoperators on Hilbert spaces, for example.
• The intersection of the row-finite and column-finite matrix rings forms a ring${\displaystyle {\mathbb{R}\mathbb{C}\mathbb{F}\mathbb{M}}_I(R)}$.
• IfR iscommutative, then M_n(R) has a structure of a*-algebra overR, where theinvolution * on M_n(R) ismatrix transposition.
• IfA is aC*-algebra, then M_n(A) is another C*-algebra. IfA is non-unital, then M_n(A) is also non-unital. By theGelfand–Naimark theorem, there exists aHilbert spaceH and an isometric *-isomorphism fromA to a norm-closed subalgebra of the algebraB(H) of continuous operators; this identifies M_n(A) with a subalgebra ofB(H^⊕n). For simplicity, if we further suppose thatH is separable andA${\displaystyle \subseteq }$B(H) is a unital C*-algebra, we can break upA into a matrix ring over a smaller C*-algebra. One can do so by fixing aprojectionp and hence its orthogonal projection 1 − p; one can identifyA with, where matrix multiplication works as intended because of the orthogonality of the projections. In order to identifyA with a matrix ring over a C*-algebra, we require thatp and 1 − p have the same "rank"; more precisely, we need thatp and 1 − p are Murray–von Neumann equivalent, i.e., there exists apartial isometryu such thatp =uu* and1 −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 fieldC ofcomplex numbers. Prior to the invention of matrix algebras,Hamilton in 1853 introduced a ring, whose elements he calledbiquaternions^[7] and modern authors would call tensors inC ⊗_RH, that was later shown to be isomorphic to M₂(C). Onebasis of M₂(C) consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by theidentity matrix and the threePauli matrices.
• A matrix ring over a field is aFrobenius 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 thering of endomorphisms of thefree rightR-module of rankn; that is,M_n(R) ≅ End_R(Rⁿ).Matrix multiplication corresponds to composition of endomorphisms.
• The ring M_n(D) over adivision ringD is anArtiniansimple ring, a special type ofsemisimple ring. The rings${\displaystyle {\mathbb{C}\mathbb{F}\mathbb{M}}_I(D)}$ and${\displaystyle {\mathbb{R}\mathbb{F}\mathbb{M}}_I(D)}$ arenot simple and not Artinian if the setI is infinite, but they are stillfull linear rings.
• TheArtin–Wedderburn theorem states that every semisimple ring is isomorphic to a finitedirect product$\prod _{i=1}^r{\mathrm{M}}_{n_i}(D_i)$, for some nonnegative integerr, positive integersn_i, and division ringsD_i.
• When we view M_n(C) as the ring of linear endomorphisms ofCⁿ, those matrices which vanish on a given subspaceV form aleft ideal. Conversely, for a given left idealI of M_n(C) the intersection ofnull spaces of all matrices inI gives a subspace ofCⁿ. Under this construction, the left ideals of M_n(C) are in bijection with the subspaces ofCⁿ.
• There is a bijection between the two-sidedideals of M_n(R) and the two-sided ideals ofR. Namely, for each idealI ofR, the set of alln ×n matrices with entries inI is an ideal of M_n(R), and each ideal of M_n(R) arises in this way. This implies that M_n(R) issimple if and only ifR is simple. Forn ≥ 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 inR. For example, the set of matrices whose columns with indices 2 throughn are all zero forms a left ideal in M_n(R).
• The previous ideal correspondence actually arises from the fact that the ringsR and M_n(R) areMorita equivalent. Roughly speaking, this means that the category of leftR-modules and the category of left M_n(R)-modules are very similar. Because of this, there is a natural bijective correspondence between theisomorphism classes of leftR-modules and left M_n(R)-modules, and between the isomorphism classes of left ideals ofR and left ideals of M_n(R). Identical statements hold for right modules and right ideals. Through Morita equivalence, M_n(R) inherits anyMorita-invariant properties ofR, such as beingsimple,Artinian,Noetherian,prime.

• IfS is asubring ofR, 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) iscommutative if and only ifn = 0,R = 0, orR iscommutative andn = 1. In fact, this is true also for the subring of upper triangular matrices. Here is an example showing two upper triangular2 × 2 matrices that do not commute, assuming 1 ≠ 0 inR:

  

  and

  

• Forn ≥ 2, the matrix ring M_n(R) over anonzero ring haszero divisors andnilpotent elements; the same holds for the ring of upper triangular matrices. An example in2 × 2 matrices would be

  

• Thecenter of M_n(R) consists of the scalar multiples of theidentity matrix,I_n, in which the scalar belongs to the center ofR.
• Theunit group of M_n(R), consisting of the invertible matrices under multiplication, is denoted GL_n(R).
• IfF is a field, then for any two matricesA andB in M_n(F), the equalityAB =I_n impliesBA =I_n. This is not true for every ringR though. A ringR whose matrix rings all have the mentioned property is known as astably finite ring (Lam 1999, p. 5).

In fact,R needs to be only asemiring for M_n(R) to be defined. In this case, M_n(R) is a semiring, called thematrix semiring. Similarly, ifR is a commutative semiring, then M_n(R) is amatrix semialgebra.

For example, ifR is theBoolean semiring (thetwo-element Boolean algebraR = {0, 1} with1 + 1 = 1),^[8] then M_n(R) is the semiring ofbinary relations on ann-element set with union as addition,composition of relations as multiplication, theempty relation (zero matrix) as the zero, and theidentity relation (identity matrix) as theunity.^[9]

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

• Algebraic structures
• Ring theory
• Matrix theory

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