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Concept
Matrix ring
Summary
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. Any subring of a matrix ring is 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.
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 any index set I, the ring of endomorphisms of the right R-module is isomorphic to the ring 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 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 .
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.
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