Intuition of the torsion submodule What is the motivation for defining the torsion submodule $M_\text{tors}$ of a module $M$? Is there an insightful way of thinking about it intuitively?
 A: The "elements of finite order"-perspective offered by Nicolas Bourbaki is a good start. Let add my two cents.
The notion of non-trivial torsion elements is one of the main things setting the study of general modules apart from the study of vector spaces. Indeed, the torsion phenomenon is non-existent for vector spaces ($\lambda v=0$ always implies either $\lambda=0$ or $v=0$) but very natural for modules (consider, for example, any quotient ring).
One can think about the torsion of a module as roughly the non-free part ($=$ the part that does not behave like a vector space). This is further supported by the fact that free modules are torsion-free. Be warned that this is only a rough analogy guiding some basic intuition and far from being precise in general.
For certain base rings, this intuition carries over to an actual structural statement. This is the case e.g. if we consider modules over $R$ a principal ideal domain. Then one can show that any (finitely generated) $R$-module $M$ decomposes as a direct sum
$$
M\cong M_\text{tors}\oplus R^r
$$
for some uniquely determined non-negative integer $r$ (its so-called rank measuring the size of its free part). Hence, there is a non-free part ($M_\text{tors}$) and a free part ($R^r$) which one can strictly separate and the torsion part is what differentiates $M$ from being free (i.e. structurally something like a vector space). One can actually further decompose the torsion part as a direct sum of quotient rings but this is not really relevant right now.

However, in general such structure theorems depend highly on the base ring $R$. In particular, there is not always the possibility of separating the torsion and torsion-free part (the quotient by its torsion submodule) making the analogy somewhat fuzzy.
There is an interesting example of this from algebraic number theory: consider as base ring $\Lambda=\mathbb Z_p[[T]]$ a ring of formal power series over the $p$-adic integers (this ring is also called the Iwasawa algebra due to its origins in Iwasawa theory). One can show -although the proof is quite involved- that any finitely generated $\Lambda$-module $M$ can  again be related to the direct sum
$$
M_\text{tors}\oplus \Lambda^r\,.
$$
This relation, however, is not by an isomorphism but along something weaker called a pseudo-isomorphism. There are certain cases where there is only a pseudo-isomorphism and no isomorphism whatsoever due to the complicated internal structure of $\Lambda$. A good example of this is $\mathfrak m=(p,T)$ (the unique maximal ideal of $\Lambda$) as $\Lambda$-module which is neither free nor torsion.
One should keep this in mind.
