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
Famous quotes containing the words matrix and/or ring:
““The matrix is God?”
“In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
“If it has limits, it isn’t omnipotent.”
“Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
—William Gibson (b. 1948)
“I started out very quiet and I beat Mr. Turgenev. Then I trained hard and I beat Mr. de Maupassant. I’ve fought two draws with Mr. Stendhal, and I think I had an edge in the last one. But nobody’s going to get me in any ring with Mr. Tolstoy unless I’m crazy or I keep getting better.”
—Ernest Hemingway (1899–1961)