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[x1,...] over a field by some ideal. A Stanley decomposition of R is a representation of R as a direct sum (of vector spaces)


 * $$ R = \bigoplus_\alpha x_\alpha k(X_\alpha) $$

where each xα is a monomial and each Xα is a finite subset of the generators.