In abstract algebra the matrix ring M(n, R) is the set of all n×n matrices over an arbitrary ring R. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real This set is itself a ring under matrix addition and multiplication. In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix
Properties
- The matrix ring M(n, R) is commutative if and only if n ≤ 1 or R is the trivial ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property A trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r
- The center of a matrix ring over R consists of the matrices which are of the form b times the identity matrix, where b belongs to the center of R. The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main
- A matrix ring over a division ring is an Artinian simple ring. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Abstract algebra, an Artinian ring is a ring that satisfies the Descending chain condition on ideals. In Abstract algebra, a simple ring is a non-zero ring that has no ideal besides the Zero ideal and itself The converse is also true and called the Artin-Wedderburn theorem. In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings.
- A matrix ring over a prime ring is a prime ring. In Abstract algebra, a non-trivial ring R is a prime ring if for any two elements a and b of R, if arb
- The matrix ring M(n, R) can be identified with the endomorphism ring End(Rn). In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object
- There is a one-to-one correspondence between the (two-sided) ideals of M(n, R) and the (two-sided) ideals of R. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Namely, for each ideal I of R, the set of all n-by-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. In Abstract algebra, a simple ring is a non-zero ring that has no ideal besides the Zero ideal and itself However, this correspondence is not one-to-one if we consider right ideals (or left ideals) rather than two-sided ideals. For example, M(n, R) has the proper nonzero right ideal consisting of all matrices whose columns are zero except for possibly the first column, but the set of all entries of matrices from this ideal is equal to R. The correspondence for left and right ideals is rather with submodules of free modules.
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