# Abstract Nonsense

## Modules (Pt. I)

Point of Post: In this post we introduce the notion of modules and their pursuant definitions.

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Motivation

We now begin discussing one of the most important and prevalent structures in modern mathematics–modules.  So, as I always say, before we start belting out definitions and theorems its important to understand precisely what modules are, why anyone would consider them, and why they are useful. So, this is what we shall now do, in parts!

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Definition

So, let’s begin by defining (roughly) what modules are. Out of all the structures in mathematics modules sit in a unique place to be easily explained. Namely, when one studies modules one usually has had enough math that the idea of a module comes quite easily. Namely, a module is just a vector space where the scalars are elements in some (not necessarily unital or commutative) ring. Said differently, if one goes back to the definition of a vector space (over a field) which everyone reading this (presumably) has seen (if not, start there!) and replaces the field $F$ with a general ring $R$ one has a (left) $R$-module!

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Why Someone Would Consider Modules

So, now lets move onto why someone would ever consider modules. This is prettily well reworded as “where have I seen modules before?” Luckily this is, once again, an extremely ‘easy’ question to answer (since modules abound in mathematics). Indeed, if one really sat down and thought about it I bet one could think of instances in their mathematics career where they wanted to do ‘linear algebra’ in a situation that was totally ‘linear algebraic’ but couldn’t because their scalars weren’t in a field! Probably the earliest memory I have of this was trying some competition math problems where I had to prove some fact about matrices in $\text{Mat}_n(\mathbb{Z})$. For example, I might try to say something like “Oh! Well, since the determinant is nonzero we know that it has an inverse in $\text{Mat}_n(\mathbb{Z})$!” or “Oh, well since the standard basis for $\mathbb{R}^n$ is also a basis for $\mathbb{Z}^n$…wait, what does basis mean here?” (although technically if one does enough group theory, it’s clear what it means)

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Of course, this is not the only case of having ‘linear algebra over rings’. Probably the second most interesting example comes from linear algebra itself. Namely, suppose we are given some $k$ vector space $V$. We know that the set $\text{End}(V)$ of endomorphisms on $V$ is an abelian group (in fact, it’s an associative algebra). Moreover, there is a natural ring which acts on $\text{End}(V)$ (although one may have never thought about it directly), namely $k[x]$. All the time in linear algebra we are taking polynomials of $T\in\text{End}(V)$ never realizing that we are doing ‘linear algebra’ when we think of $\text{End}(V)$ as a $k[x]$-module. To elucidate (very slightly) that this isn’t just semantic naming of things one can recall the importance of the minimal polynomial $m_T(x)$ for some $T\in\text{End}(V)$ in the study of $T$. Well, it turns out that the set of all $k[x]$ annihilating $T$ is an ideal. But, we know that $k[x]$ is a PID and so this ideal is generated by some element of $k[x]$…I’ll give you three guesses which one (hint: it’s $m_T(x)$).

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So, we have discussed two examples where the idea of generalizing vector spaces to modules led to some interesting results. We’d now like to point out two examples which are in a different spirit. Namely, modules subsume two giant classes of objects which I have previously studied. First, it’s not hard to imagine (analogizing with vector spaces) that every ring is a module over itself. Thus, modules can (in a very real sense) be seen as a generalization of ring theory, The second thing to notice is that if one is a given some abelian group $A$ there is a natural way to turn $A$ into $\mathbb{Z}$-module. How? Well, as funny as its sounds its just the $n$-fold summation we never think about using. Namely, $z\cdot a$ for $a\in A$ is just the $|z|$-fold sum of $a$ adjusted to be plus or minus depending upon the sign of $z$. Thus, we see that module theory naturally generalizes abelian groups (often the infinite ones, but not always).

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Uses

So, where are we going to find modules useful? By this, I mean besides the obvious appeal of being able to due more general ‘linear algebra’.  Well, there are two powerful theorems which we shall eventually prove using module theory that one has mostly likely already seen: the fundamental theorem for finitely generated abelian groups and the Jordan decomposition theorem (gotten by studying the action of $\mathbb{Z}$ on abelian groups and $k[x]$ on $k$-vector spaces respectively). That said, we shall (hopefully) see that modules pop up everywhere in some of the more advanced subjects. Indeed, modules hold a key position in algebraic topology and algebraic geometry. Moreover, representation theory is almost entirely (secretly, the way I have done it on this blog, I hope to ‘redo’ it in some sense module theoretically soon enough) the study of the interaction of groups and modules. The same could be said about group cohomology. We shall also see (I am doing an independent study in this next term, so this should be clear soon enough) that the category of $R$-modules shall serve as a prime example and motivation  in our study of category theory.

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Modules

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So, there are two equivalent ways we can think about modules. Let’s first think about modules in the way I stated in the motivation, as “vector spaces” over rings. Namely, let $M$ be an abelian group, $R$ a ring, and $\mu:R\times M\to M$ be a given map. Then, the triple $(M,R,\mu)$ is called a left R-module provided the following axioms hold for all $r,s\in R$ and $x,y\in M$

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\displaystyle \begin{aligned}&\mathbf{(1)}\quad\mu(r,x+y)=\mu(r,x)+\mu(r,y)\\ &\mathbf{(2)}\quad \mu(r+s,x)=\mu(r,x)+\mu(s,x)\\ &\mathbf{(3)}\quad \mu(rs,x)=\mu(r,\mu(s,x))\end{aligned}

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The ring $R$ is called the ground ring or base ring. and $\mu$ is called the multiplication map. As always, we omit $\mu$ for concatenation so that, for example, axiom $\mathbf{(1)}$ would read $r(x+y)=rx+ry$. If in addition we have that $R$ is unital we say that $M$ is a unital left $R$-module if in addition to the above axioms we have that $1_Rx=x$ for all $x\in M$. In the majority of cases we shall be assuming that our rings and modules are unital and so, if unsure, assume it’s unital.

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So, this clearly mirrors the definition of a vector space in the sense that a unital left $k$-module when $k$ is a field is nothing more than a vector space. So, what is the other way to look at left modules? Well, there is an admittedly different way to motivate modules which while, in a sense, is “better” is less intuitive. To be particular, it’s doubtless that anyone associated with any kind of group theory (e.g. finite group theory) is well-acquainted with how much information about a group is contained in its actions on objects (e.g. Sylow’s theorems). Consequently, it makes sense that to understand rings we should perhaps look at the actions of rings on sets. So, what exactly should this mean? Well, the ‘obvious’ definition (modeling off of the definition of group actions)  is sort of ambiguous since we’d like to (in a perfect world where everything generalizes nicely) just define a ring action to be a homomorphism $R\to \text{Sym}(X)$ for some set $X$, right? But, there’s a problem…what kind of homomorphism? Clearly we can’t exclaim “ring!” as would be easy since there is no natural way (let alone a predefined one) to turn $\text{Sym}(X)$ into a ring. That said, if $X$ had more structure than just some set, if it was something richer, perhaps we could find a set analogous to $\text{Sym}(X)$ (i.e. a set of mappings on a fixed object) which would do the trick. ::cue perfectly timed realization:: Aha! There is something that fits the bill perfectly. In particular, if $X$ happened to be an abelian group then we know that set $\text{End}(X)$ is naturally a ring under composition and point-wise addition. Thus, what if we defined a ring action of a ring $R$ on the abelian group $X$ to be a ring homomorphism $\phi:R\to \text{End}(X)$, which should be a unital homomorphism when $R$ is. One can quickly check then that if one has a ring action $\phi:R\to\text{End}(X)$ then the map $\mu:R\times X\to X:(r,x)\mapsto \phi_r(x)$ defines a natural left $R$-module structure on $X$, and conversely if $M$ is a left $R$-module with multiplication map $\mu$ then one can quickly check that the map $\phi:R\to \text{End}(M)$ given by $\phi(r)=\mu(r,-)$ is a ring action. Thus, studying modules is really the same pursuit as studying ring actions on abelian groups.

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References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

October 27, 2011 -

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