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Matrix ring

"Matrix algebra" redirects here. For the algebraic theory of matrices, see Matrix (mathematics) and Linear algebra.

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.[1] The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R)[2][3][4][5] (alternative notations: Matn(R)[3] and Rn×n[6]). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again 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.

  1. ^ Lam (1999), Theorem 3.1
  2. ^ Lam (2001).
  3. ^ a b Lang (2005), V.§3
  4. ^ Serre (2006), p. 3
  5. ^ Serre (1979), p. 158
  6. ^ Artin (2018), Example 3.3.6(a)

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