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 fact about::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.