Regular sequence on a module

Definition
Let $$R$$ be a commutative unital ring, $$M$$ a $$R$$-module, and $$x_1, x_2, \ldots, x_n$$ be a sequence of elements in $$R$$. We say that the $$x_i$$s form a regular sequence on $$M$$ if the following two conditions hold:


 * $$(x_1,x_2,\ldots,x_n)M \ne M$$
 * For $$1 \le i \le n$$, $$x_i$$ is a nonzerodivisor on $$M/(x_1,x_2,\ldots,x_{i-1})M$$

When no module is specified, we assume the module to be $$R$$ itself.

Facts

 * If $$R$$ is a Noetherian local ring and $$x_1, x_2, \ldots, x_n$$ form a regular sequence in its unique maximal ideal, then any permutation of the $$x_i$$s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular.
 * If $$R$$ is a graded ring, and $$x_1, x_2, \ldots, x_n$$ form a regular sequence and all the $$x_i$$s are homogeneous elements, then any permutation of the $$x_i$$s is also a regular sequence.
 * If $$x_1, x_2, \ldots, x_d$$ are a regular sequence on a module $$M$$ over a Noetherian local ring, then the difference of degrees of the Hilbert-Samuel polynomial for $$M$$ and for $$M/(x_1,x_2,\ldots,x_d)$$ is at least $$d$$.