---- > [!definition] Definition. ([[direct sum of modules]]) > Let $R$ be a [[ring]] and let $M$ and $N$ be $R$-[[module|modules]]. We can endow the [[direct sum of abelian groups|direct sum]] of [[abelian group|abelian groups]], $M \oplus N$, with $R$-[[module]] structure via the [[module|action]] $\big( r, (m,n) \big) \mapsto (rm, rn).$ > This defines the **direct sum of $R$-modules** $M \oplus N$. > > [!generalization] Generalizing. > > [[direct sum of modules#^definition-2|See below.]] ^definition > [!note] Motivation. > In addition to being intuitive, this definition is a good one because > > **1.** $M \oplus N$ is at once a [[categorical product]] and [[categorical coproduct]] in $R\text{-}\mathsf{Mod}$; [^1] > > > **2.** This is the *only* way to place module structure on the [[cartesian product]] $M \times N$ such that the canonical projection set-functions $\begin{align} > \begin{array} \\ > \pi_{M}:M \times N \to M \\ > (m, n) \mapsto m > \end{array} \text{ and } \begin{array} > \ \pi_{N}: M \times N \to N \\ > (m, n) \mapsto n > \end{array} > \end{align}$are [[linear map|module homomorphisms]]. > > **3.** This is the *only* way to place module structure on the [[cartesian product]] $M \times N$ such that the canonical inclusion set-functions $\begin{align} > \begin{array} \\ > \iota_{M}:M \to M \times N\\ > m \mapsto (m, 0) > \end{array} \text{ and } \begin{array} > \ \iota_{N}: N \to M \times N \\ > n \mapsto (0, n) > \end{array} > \end{align}$are [[linear map|module homomorphisms]]. ^justification [^1]: This has the consequence that any $R$-[[linear map|linear map]] $\varphi:M_{1}\oplus\dots \oplus M_{n} \to N$ just equals $\sum_{i=1}^{n} \varphi_{i}$ for some collection $\varphi_{i}:M_{i} \to N$ > [!definition] Definition. (Direct Sum of a Family of Modules) > Let $\{ N_{j} \}_{j \in J}$ be an indexed family of $R$-[[module|modules]]. As expected, endowing the [[direct sum of abelian groups|direct sum of the underlying abelian groups]] $\bigoplus_{j \in J}N_{j}=\big{\{} \text{Indexed tuples } (n_{j})_{j \in J} \text{ s.t. } n_{j} \in N_{j} \text{ and } \{ j \in J :h_{j} \neq 0_{j}\} \text{ is finite} \big{\}}$ > with the obvious $R$-module structure $\big( r , (n_{j})_{j \in J} \big) \mapsto (rn_{j})_{j \in J}$ > yields an $R$-[[module]] which satisfies the [[universal property]] for general [[categorical coproduct|coproducts]] in $R$-$\mathsf{Mod}$. > > > [!specialization] Specializing. (Direct Sum Power of an $R$-Module) > > When $N$ is an $R$-[[module]] such that $N_{j}=N$ for all $j \in J$, the set $\bigoplus_{j \in J}N_{j}=N ^{\oplus J}$ can be viewed as the set $\text{Hom}_{\mathsf{Set}}(J, N)$ of functions $J \xrightarrow{f} H$ which 'satisfy the finiteness condition': $\coprod_{j \in J}N = N^{\oplus J}=\big\{ \text{Functions } f: J \to N \text{ s.t. } \{ j \in J: f(j) \neq 0 \} \text{ is finite} \big\}.$ > > > ^definition-2 > [!justification] > To begin, we should specify what are the coprojections $\iota_{j}$. $\iota_{j}(n_{j})$ equals the indexed tuple $(n_{j})_{j \in J}$ whose $i^{th}$ element is $\begin{cases} > 0 & i \neq j; \\ > n_{j} & i=j. > \end{cases}$ > Then, given an $R$-[[module]] $L$, and maps $\{ N_{j} \xrightarrow{\varphi_{j}} L \}_{j \in J}$ the [[universal property|universal function]] $\sigma$ can only have one definition... Commutativity ($\sigma \circ \iota_{j}(n_{j})=\varphi_{j}(n_{j})$ for all $j \in J$ and $n_{j} \in N_{j}$) and the fact that $\sigma$ should be a [[linear map]] enforce > $\sigma\big( (n_{j})_{j \in J} \big)=\sigma\left( \sum_{j \in J} \iota_{j}(n_{j}) \right)=\sum_{j \in J} \sigma \circ \iota_{j}(n_{j})=\sum_{j \in J} \varphi_{j}(n_{j}).$This is uniqueness (and it is really just a rehash of the [[direct sum of abelian groups]] discussion); existence (i.e. that $\sigma$ is indeed a [[linear map]]) is obvious. ---- #### > [!proof] Proof of Remark. **1 - Product.** Let $P$ be an $R$-[[module]], and let $\varphi_{M}:P \to M$ and $\varphi_{N}:P \to N$ be two $R$-[[module]] [[linear map|homomorphisms]]. The [[direct product of groups]] discussion means we just have to check that the function $\sigma=\varphi_{M} \times \varphi_{N}: P \to M \oplus N, p \mapsto (\varphi_{M}(p), \varphi_{N}(p))$ respects the ring action. Indeed, $\sigma(rp)=\big( \varphi_{M}(rp), \varphi_{N}(rp) \big)=\big(r \varphi_{M}(p) , r \varphi_{N}(p)\big)=r\sigma(p).$ **1 - Coproduct.** Mirror the above. With $P$ an $R$-[[module]], let $\psi_{M}: M \to P$ and $\psi_{N}:N \to P$ be two $R$-[[linear map|module homomorphisms]]. In light of [[direct products and coproducts align in the category of abelian groups]] it is enough to check that the function $\sigma': \psi_{M} \oplus \psi_{N}: M \oplus N \to P, (m,n) \mapsto \psi_{M}(m)+ \psi_{N}(n)$ respects the ring action. Indeed, $\begin{align} \sigma'\big( r (m,n) \big)= & \sigma'\big( (rm, rn) \big) \\ = & \psi_{M}(rm) + \psi_{N}(rn) \\ = & r \psi_{M}(m) + r \psi_{N}(n) \[](direct%20products%20and%20(finite?)%20coproducts%20align%20in%20the%20category%20of%20abelian%20groups.md)) \big). \end{align}$ > **2.** From [[direct product of groups]], we already know that $M \oplus N$ has the unique *[[group]]* structure under which the projections are [[group homomorphism|group homomorphisms]]. Since the projections are obviously [[linear map|module homomorphisms]] ($\pi_{M}(r(m,n))=\pi_{M}(rm, rn)=rm=r \pi_{M}(m,n)$), we are done. > **3.** Entirely analogous to (2) above. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```