Is a regular sequence ordered? A regular sequence is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring
$$ \frac R {\langle r_1, r_2, \cdots, r_{i-1} \rangle}.$$

Does the order of the $r_i$'s matter? That is, is any permutation of a regular sequence regular?


Here is a general result for when any permutations of elements of a regular sequence forms a regular sequence:
Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. If $x_1,...,x_n$ be an $M$-sequence s.t. $x_i \in J(A)$ for $1 \leq i \leq n$, where $J(A)$ is the Jacobson radical of $A$, then any permutation of $x_1,...,x_n$ becomes an $M$-sequence.
