Permutation of regular sequence is not necessarily regular
The following are true:
- If is a Noetherian local ring, and the regular sequence comprises elements in the unique maximal ideal, then any permutation of it is regular.
- If is a graded ring, and all the elements in the regular sequence are homogeneous elements, then every permutation is regular.
- Any permutation of a regular sequence of length one is regular (obviously)
- For a unique factorization domain, and when the module is the ring itself, a sequence of length two is regular, if and only if the two elements are relatively prime. Thus, any permutation of a regular sequence of length two is regular.
- For a principal ideal domain, when the module is the ring itself, there cannot exist regular sequences of length more than two, so any permutation of a regular sequence is regular.
One example uses the polynomial ring over a field. Consider the sequences:
The first one of these is regular, while the second clearly isn't: in the second sequence, the term is actually equal to zero modulo the first two terms.