# Abstract Nonsense

## Internal Direct Sum of Modules

Point of Post: In this post we discuss the notion of the internal directs sum of modules, giving the usual characterizations, etc. We also give a standard application to short exact sequences.

$\text{ }$

Motivation

We have seen that the category of modules has notions of coproducts, namely given a set $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ of modules, we can form the module $\displaystyle \bigoplus_{\alpha\in\mathcal{A}}M_\alpha$ which is the submodule of the product $\displaystyle \prod_{\alpha\in\mathcal{A}}M_\alpha$ of $\mathcal{A}$-tuples with finite support. That said, from experience dealing with vector spaces, we often see that coproducts sometimes come up more naturally as “internal direct sums”. Namely, a module $M$ may be able to be written as the sum of submodules $\{M_\alpha\}$ in a “unique way” (i.e. unique representation). In this case, $M$ will be isomorphic to the coproduct over the set $\{M_\alpha\}$ of modules, and so we don’t really get “new theory”. That said, often times we want to work with actual equalities: “$M$ is the internal direct sum” opposed to “$M$ is isomorphic to the coproduct”.

$\text{ }$

Internal Direct Sum of Modules

$\text{ }$

Let $R$ be a ring and $M$ a left $R$-module. Suppose that$\{M_\alpha\}_{\alpha\in\mathcal{A}}$ is a family of submodules of $M$. We clearly have, for each $\alpha\in\mathcal{A}$, the inclusion maps $\iota_\alpha:M_\alpha\to M$. We see then by definition of coproduct that we are afforded a guaranteed map $\displaystyle f:\bigoplus_{\alpha\in\mathcal{A}}M_\alpha\to M$. We say that $M$ is the internal direct sum of $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ if $f$ is an isomorphism. Since, in an obvious sense, internal direct sums and coproducts are the same thing we shall use the same symbol $\oplus$ to denote both. While it should be clear from context which we actually mean, the tipoff shall often come in the form of equality or isomorphism. For example, if one writes $\displaystyle M=\bigoplus_{\alpha\in\mathcal{A}}M_\alpha$ then its clear that we mean that $M$ is the internal direct sum of this family of modules (which are submodules).

$\text{ }$

Of course, while the above is a nice, succinct way of putting internal direct sums, it is often not how one actually thinks about the concept. No, really one often thinks about a module as being an internal direct sum of a family if the following characterization holds:

$\text{ }$

Theorem: Let $M$ be a left $R$-module and $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ a family of submodules. Then, $M$ is the internal direct sum of $\{M_\alpha\}$ if and only if every element of $M$ can be written uniquely as a sum $\displaystyle \sum_{\alpha\in\mathcal{A}}m_\alpha$ with $m_\alpha\in M_\alpha$ with all but finitely many $m_\alpha=0$.

Proof: The fact that $\displaystyle \sum_{\alpha\in\mathcal{A}}M_\alpha=M$ is equivalent to the statement that the canonical map $\displaystyle f:\bigoplus_{\alpha\in\mathcal{A}}M_\alpha\to M$ is surjective, and the uniqueness of this representation is equivalent to the injectivity of $f$. $\blacksquare$

$\text{ }$

For the case when we are dealing with only two submodules, this takes the following particularly nice form:

$\text{ }$

Theorem: Let $R$ be a ring and $M$ a left $R$-module. Then, if $N,L\leqslant M$ then $M=L\oplus N$ if and only if $M=L+N$ and $L\cap N$ is trivial.

Proof: Since we have a priori that $M=L+N$ it suffices to show that uniqueness of representation is equivalent the triviality of $N\cap L$. Suppose first that we have uniqueness of representation. If $x\in M\cap N$, then we can write it as $x=0+x$ where we are thinking about $0\in L$ and $x\in N$ and $x=x+0$ where $x\in L$ and $0\in N$. From uniqueness we may conclude that $x=0$. Conversely, if $L\cap N$ is trivial and if $\ell+n=\ell'+n'$ then $\ell-\ell'=n'-n$ and so clearly $\ell-\ell',n'-n\in N\cap L$ and so $\ell-\ell'=n'-n=0$ and so uniqueness of representation follows. $\blacksquare$

$\text{ }$

In fact, it’s not hard to see that this generalizes to the following:

$\text{ }$

Theorem: Let $M$ be a left $R$-module and $\left\{M_\alpha\right\}_{\alpha\in\mathcal{A}}$ a family of submodules. Then, $M$ is the internal direct sum of $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ if and only if $\displaystyle \sum_{\alpha\in\mathcal{A}}M_\alpha=M$ and $\displaystyle M_\alpha\cap\sum_{\beta\ne\alpha}M_\beta$ is trivial for each $\alpha\in\mathcal{A}$.

$\text{ }$

$\text{ }$

Relation to Short Exact Sequences

$\text{ }$

We can use this notion of internal direct sums to be able to write the central module of a split short exact sequence in a nice way, in terms of the image and kernel of certain maps. First we note the following:

$\text{ }$

Theorem: Let $R$ be a ring and $M,N$ left $R$-modules. If $f:M\to N$ and $g:N\to M$ are $R$-maps such that $f\circ g=\text{id}_N$ then $M=\text{im }g\oplus\ker f$.

Proof: By the above theorems it suffices to show that $\text{im }g\cap\ker f$ is trivial and $\text{im }g+\ker f=M$. To see the first, suppose that $x\in\text{im }g\cap\ker f$. Since $x\in\text{im }g$ we can find $n\in N$ with $x=g(n)$. But, note then that says that $f(g(n))=f(x)$ but, since $x\in\ker f$ we see that $f(x)=0$ and since $f\circ g=\text{id}_N$ we see that $f(g(n))=n$ and so $n=0$. Thus, $x=g(n)=g(0)=0$. Since $x$ was arbitrary the triviality of $\text{im }g\cap\ker f$ follows. To see that $M=\text{im }g+\ker f$ we let $x\in M$ be arbitrary. Note then that $x=(x-g(f(x)))+g(f(x))$. Clearly $g(f(x))\in\text{im }g$ and since $f(x-g(f(x))=f(x)-f(g(f(x))=f(x)-f(x)=0$ we see that $x-g(f(x))\in\ker f$ and so clearly $x\in \text{im }g+\ker f$. Since $x$ was arbitrary the conclusion follows. $\blacksquare$

$\text{ }$

Of course, applying this in conjunction with the splitting lemma gives us the following:

$\text{ }$

Theorem: Let $0\to A\xrightarrow{h}B\xrightarrow{f}C\to0$ be a split short exact sequence. If $g:C\to B$ is the guaranteed backmap for $f$ then $B\cong \ker f\oplus\text{im }g$.

$\text{ }$

$\text{ }$

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] Blyth, T. S. Module Theory. Clarendon, 1990. Print.