Permutation of regular sequence is not necessarily regular

Statement

Let $R$ be a commutative unital ring, $M$ a $R$-module, and $x_{1},x_{2},\ldots ,x_{n}$ a regular sequence in $R$, for the module $M$. Then, it is not necessarily true that every permutation of the $x_{i}$s is regular.

Partial truth

The following are true:

• If $R$ is a Noetherian local ring, and the regular sequence comprises elements in the unique maximal ideal, then any permutation of it is regular.
• If $R$ 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 $k[x,y,z]$ over a field. Consider the sequences:

$xy,xz,y-1$

versus:

$xy,y-1,xz$

The first one of these is regular, while the second clearly isn't: in the second sequence, the term $xz$ is actually equal to zero modulo the first two terms.