Properties
Let T be a regular chain in the polynomial ring R.
- The saturated ideal sat(T) is an unmixed ideal with dimension n − |T|.
- A regular chain holds a strong elimination property in the sense that:
- .
- A polynomial p is in sat(T) if and only if p is pseudo-reduced to zero by T, that is,
- .
- Hence the membership test for sat(T) is algorithmic.
- A polynomial p is a zero-divisor modulo sat(T) if and only if and .
- Hence the regularity test for sat(T) is algorithmic.
- Given a prime ideal P, there exists a regular chain C such that P = sat(C).
- If the first element of a regular chain C is an irreducible polynomial and the others are linear in their main variable, then sat(C) is a prime ideal.
- Conversely, if P is a prime ideal, then, after almost all linear changes of variables, there exists a regular chain C of the preceding shape such that P = sat(C).
- A triangular set is a regular chain if and only if it is a Ritt characteristic set of its saturated ideal.
Read more about this topic: Regular Chain
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