What is the usefulness of $M[X]$? Given a commutative ring with unity $R$ and a $R$-module $M$, it can be defined the $R[X]$-module (here $X$ is an indeterminate) $M[X]$ as the set the formal expressions $\sum_{k=0}^nm_kX^k$, with $m_k\in M$, endowed with the natural sum and scalar product by elements of $R[X]$.
As $R$-module, $M[X]$ is not very interesting: it is just the direct sum $M^{\oplus\mathbb N}$. I want to know if the extra structure of $M[X]$ as $R[X]$-module gives some relevant information about the $R$-module $M$.
 A: This is a very natural $R[X]$-module, in fact the "universal one" over the $R$-module $M$ in the sense that we have an isomorphism of $R[X]$-modules
$$M[X] \cong M \otimes_R R[X].$$
This also offers a geometric description: If $S$ is a scheme (in the above case it's $S=\mathrm{Spec}(R)$) and $M$ is some $\mathcal{O}_S$-module, then $M[X]$ is the pullback of $M$ under the canonical projection morphism $\mathbb{A}^1_S \to S$. For a smooth variety $S$ one knows that this operation induces an isomorphism on the $K$-groups, so that these are "homotopy invariant". This operation is useful.
A: Here's the one application of $M[X]$ that I know:
Given your setup above, let $x_1, ..., x_n$ be a sequence of elements in $R$, and $I = (x_1,...,x_n)$ the ideal they generate. Then there is a natural surjection $(M/IM)[X_1,...,X_n] \twoheadrightarrow \operatorname{gr}_I(M)$, sending $X_i \mapsto \overline{x_i}$. In the case that this is an isomorphism (and neither side is $0$), $x_1,.., x_n$ are said to be $M$-quasi-regular. Essentially, this means that the elements of the original sequence act as indeterminates on $\operatorname{gr}_I(M)$.
This is closely related to the important notion of a regular sequence on $M$: regular sequences are always quasi-regular, and in the local or graded cases the converse holds. As one application, this gives a proof that in the local case, regular sequences are permutable, since quasi-regular sequences are patently so.
