|
In abstract algebra, a matrix ring is any collection of matrices over some ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of matrices with entries from ''R'' is a matrix ring denoted M''n''(''R''), as well as some subsets of infinite matrices which form infinite matrix rings. Any subring of a matrix ring is a matrix ring. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra, and may be called a matrix algebra. For this case, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''Mr'' is the matrix ''M'' with each of its entries multiplied by ''r''. This article assumes that ''R'' is an associative ring with a unit , although matrix rings can be formed over rings without unity. == Examples == * The set of all matrices over an arbitrary ring ''R'', denoted M''n''(''R''). This is usually referred to as the "full ring of ''n''-by-''n'' matrices". These matrices represent endomorphisms of the free module ''R''''n''. * The set of all upper (or set of all lower) triangular matrices over a ring. * If ''R'' is any ring with unity, then the ring of endomorphisms of as a right ''R''-module is isomorphic to the ring of column finite matrices whose entries are indexed by , and whose columns each contain only finitely many nonzero entries. The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the row finite matrices whose rows each only have finitely many nonzero entries. * If ''R'' is a normed ring, 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 and column finite matrix rings also forms a ring, which can be denoted by . * The algebra M2(R) of 2 × 2 real matrices is a simple example of a non-commutative associative algebra. Like the quaternions, it has dimension 4 over R, but unlike the quaternions, it has zero divisors, as can be seen from the following product of the matrix units: , hence it is not a division ring. Its invertible elements are nonsingular matrices and they form a group, the general linear group . *If ''R'' is commutative, the matrix ring has a structure of a *-algebra over ''R'', where the involution * on M''n''(''R'') is the matrix transposition. * Complex matrix algebras M''n''(C) are, up to isomorphism, the only simple associative algebras over the field C of complex numbers. For , the matrix algebra M''2''(C) plays an important role in the theory of angular momentum. It has an alternative basis given by the identity matrix and the three Pauli matrices. M''2''(C) was the scene of early abstract algebra in the form of biquaternions. * A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Matrix ring」の詳細全文を読む スポンサード リンク
|