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.




Matrix semiring

In fact, R needs to be only a semiring for Mn(R) to be defined. In this case, Mn(R) is a semiring, called the matrix semiring. Similarly, if R is a commutative semiring, then Mn(R) is a matrix semialgebra.

For example, if R is the Boolean semiring (the two-element Boolean algebra R = {0, 1} with 1 + 1 = 1),[8] then Mn(R) is the semiring of binary relations on an n-element set with union as addition, composition of relations as multiplication, the empty relation (zero matrix) as the zero, and the identity relation (identity matrix) as the unity.[9]

See also


  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)
  7. ^ Lecture VII of Sir William Rowan Hamilton (1853) Lectures on Quaternions, Hodges and Smith
  8. ^ Droste & Kuich (2009), p. 7
  9. ^ Droste & Kuich (2009), p. 8


  • Artin (2018), Algebra, Pearson
  • Droste, M.; Kuich, W (2009), "Semirings and Formal Power Series", Handbook of Weighted Automata, Monographs in Theoretical Computer Science. An EATCS Series, pp. 3–28, doi:10.1007/978-3-642-01492-5_1, ISBN 978-3-642-01491-8
  • Lam, T. Y. (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5
  • Lam (2001), A first course on noncommutative rings (2nd ed.), Springer
  • Lang (2005), Undergraduate algebra, Springer
  • Serre (1979), Local fields, Springer
  • Serre (2006), Lie algebras and Lie groups (2nd ed.), Springer, corrected 5th printing