Stanley decomposition

In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by .
Definition.
Suppose that a ring "R" is a quotient of a polynomial ring "k"["x"1...] over a field by some ideal. A Stanley decomposition of "R" is a representation of "R" as a direct sum (of vector spaces)
where each "x""α" is a monomial and each "X""α" is a finite subset of the generators.