In mathematics, a Sylvester domain, named after James Joseph Sylvester

James Joseph Sylvester

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...

  by , is a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

in which Sylvester's law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R, then

ρ(AB) ≥ ρ(A) + ρ(B) – n

where ρ is the inner rank of a matrix. The inner rank of an m by n matrix is the smallest integer r such that the matrix is a product of an m by r matrix and an r by n matrix.
showed that fields satisfy Sylvester's law of nullity and are therefore Sylvester domains.