In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings. Any subrings of these matrix rings are also called matrix rings.
In the case when R is a commutative ring, then the matrix ring Mn(R) is an associative algebra which may be called a matrix algebra. In this situation, 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.
It is assumed throughout that R is an associative ring with a unit 1 ≠ 0, although matrix rings can be formed over rings without unity.
Read more about Matrix Ring: Examples, Structure, Properties
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