Ideal Saturation

The ideal saturation of an ideal with respect to an ideal in a commutative ring with an identity element is the ideal

(1)

where the colon denotes an ideal quotient. Its members are the elements for which for some positive integer . The quotients form an increasing chain, which stabilizes when is a Noetherian ring. For , the saturation is written and the condition becomes . The ideal is saturated with respect to when (Jensen 2014, pp. 67-68).

In the polynomial ring over a field , saturation by a polynomial can be computed as an elimination ideal,

(2)

where is the ideal extension of to and is a new variable. The Rabinowitsch trick enforces by the equation (Trott 2006). A Gröbner basis with lexicographic order placing before all the computes this intersection.

When is algebraically closed, the corresponding zero sets satisfy

(3)

where the bar denotes closure in the Zariski topology. Thus the resulting zero set can still contain points where . For example, in , . The line is the closure of its part with , and contains the origin. Consequently, exact solution sets with a nonvanishing condition require retaining that condition after elimination.