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 n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). 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 Mn(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.
Examples The set of all n × n square matrices over R, denoted Mn(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 diagonal matrices over R. This subalgebra of Mn(R) is isomorphic to the direct product of n copies of R. For G any permutation group, the group ring R [ G ] {\displaystyle R[G]}
For any index set I, the ring of endomorphisms of the right R-module M = ⨁ i ∈ I R {\textstyle M=\bigoplus _{i\in I}R} is isomorphic to the ring C F M I ( R ) {\displaystyle \mathbb {CFM} _{I}(R)} 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 R F M I ( R ) {\displaystyle \mathbb {RFM} _{I}(R)} of row finite matrices. 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-finite and column-finite matrix rings forms a ring R C F M I ( R ) {\displaystyle \mathbb {RCFM} _{I}(R)} . If R is commutative, then Mn(R) has a structure of a *-algebra over R, where the involution * on Mn(R) is matrix transposition. If A is a C*-algebra, then Mn(A) is another C*-algebra. If A is non-unital, then Mn(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 Mn(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 ( p A p p A ( 1 − p ) ( 1 − p ) A p ( 1 − p ) A ( 1 − p ) ) {\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 Mn(C) are, up to isomorphism, the only finite-dimensional simple associative algebras over the field C of complex numbers. Prior to the invention of matrix algebras, Hamilton in 1853 introduced a ring, whose elements he called biquaternions 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).
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