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...

n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn. If the ring R is commutative, that is...

algebra, a generic matrix ring is a sort of a universal matrix ring. We denote by F n {\displaystyle F_{n}} a generic matrix ring of size n with variables...

is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have...

operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative. If G is an...

used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more...

unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n × n {\displaystyle n\times n} matrices. In some...

columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number...

In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule. If T {\displaystyle T} and U...

of noncommutative rings: The matrix ring of n-by-n matrices over the real numbers, where n > 1 Hamilton's quaternions Any group ring constructed from a...

commutative ring R is R itself. The center of a skew-field is a field. The center of the (full) matrix ring with entries in a commutative ring R consists...

semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings...

is a product of matrix rings of division rings, it is implicitly assumed that "matrix rings" refer to "full matrix rings". Every ring is (isomorphic to)...

n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are...

this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal...

the ring of real square matrices of order n is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication...

half-integers, is a noncommutative domain. A matrix ring Mn(R) for n ≥ 2 is never a domain: if R is nonzero, such a matrix ring has nonzero zero divisors and even...

standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication. A commutative algebra is...

invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over...

The Matrix is a 1999 science fiction action film written and directed by the Wachowskis. It is the first installment in the Matrix film series, starring...

the Cayley transform, which was originally defined on the 3 × 3 real matrix ring. Linear fractional transformations are widely used in various areas of...

{\displaystyle m\times n} matrices with entries in a ring K forms a ring K m , n {\displaystyle K_{m,n}} . The zero matrix 0 K m , n {\displaystyle 0_{K_{m,n}}\,}...

ring is a prime ring, and more generally: every left or right primitive ring is a prime ring. Any matrix ring over an integral domain is a prime ring...

examples is that of integral group rings. Some examples of orders are: If A {\displaystyle A} is the matrix ring M n ( K ) {\displaystyle M_{n}(K)} over...

Wedderburn–Artin theorem states that a simple Artinian ring A is a matrix ring over a division ring. Indeed, let I be a minimal (nonzero) right ideal of...

A} must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings. Examples of 2 × 2 {\displaystyle...

This is a list of characters from The Matrix franchise universe. Many of the characters listed here have names reflecting certain aspects of them, such...

of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division...

polynomial equation which holds for all matrices A in a specified matrix ring Mn(R). Matrix polynomials are often demonstrated in undergraduate linear algebra...