# Permutation of regular sequence is not necessarily regular

From Commalg

## Statement

Let be a commutative unital ring, a -module, and a regular sequence in , for the module . Then, it is *not* necessarily true that every permutation of the s is regular.

## Partial truth

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.

## Example

One example uses the polynomial ring over a field. Consider the sequences:

versus:

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.