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

Structure

Properties

Matrix semiring

In fact, R needs to be only a semiring for Mn(R) to be defined. In this case, Mn(R) is a semiring, called the matrix semiring. Similarly,**** if**** R is**** a commutative semiring****,**** then**** Mn****(R)**** is**** a . For example, if R is the Boolean semiring (the two-element Boolean algebra R = with 1 + 1 = 1), then Mn(R) is the semiring of binary relations on an n-element set with union as addition, composition of relations as multiplication, the empty relation (zero matrix) as the zero, and the identity relation (identity matrix) as the unity.

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