# Sylvester domain

In mathematics, a **Sylvester domain**, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring 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.

Sylvester (1884) showed that fields satisfy Sylvester's law of nullity and are therefore Sylvester domains.

## References

- Dicks, Warren; Sontag, Eduardo D. (1978), "Sylvester domains",
*Journal of Pure and Applied Algebra*,**13**(3): 243–275, doi:10.1016/0022-4049(78)90011-7, ISSN 0022-4049, MR 509164 - Sylvester, James Joseph (1884), "On involutants and other allied species of invariants to matrix systems",
*Johns Hopkins university circulars*,**III**: 9–12, 34–35, Reprinted in collected papers volume IV, paper 15

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