direct sum of modules as categorical coproduct

1. Proposition

Let \(R\) be a ring and \(M_i\) be a family of modules. Then the direct sum

\begin{align*} \oplus_{i \in I} M_i \end{align*}

with the inclusion

\begin{align*} \iota: M_i \rightarrow& \oplus_{i \in I} M_i \\ m \mapsto& (m) \otimes (0)_{j \neq i} \end{align*}

is the categorical coproduct of category RMod

2. Proof

Suppose there exists a module \(N\) and module-homomorphisms \(\varphi_i: M_i \rightarrow N\). Then

\begin{align*} f: \oplus_{i \in I} M_i \rightarrow N (m_i) \mapsto& \sum_{i \in I} f_i(m_i) \end{align*}

is the unique map

2.1. welldefined map

Note that by construction of \(\bigoplus_{i \in I} M_i\), we get \(m_i = 0\) for cofinitely many \(i \in I\) Hence the sum is welldefined

2.2. module-homomorphism

follows from

\begin{align*} f(m_i + r \cdot n_i) =& \sum_{i \in I} f_i(m_i + r \cdot n_i) \\ =& \sum_{i \in I} f_i(m_i) + f_i(r \cdot n_i) \\ =& \sum_{i \in I} f_i(m_i) + r \cdot f_i(n_i) \\ =& \sum_{i \in I} f_i(m_i) + r \cdot \sum_{i \in I} r \cdot f_i(n_i) \\ =& f(m_i) + r \cdot \sum_{i \in I} f_i(n_i) \\ =& f(m_i) + r \cdot f(n_i) \end{align*}

2.3. commuting

Note that

\begin{align*} f\circ \iota(m) =& f((m)_{i} \times (0)_{j \neq i}) \\ =& \sum_{i \in I} f_i(m_i \times (0)_{j \neq i}) \\ =& f_i(m) + \sum_{j \neq i} f_j(0) \\ =& f_i(m) \end{align*}

2.4. uniqueness

Note that this map is generated by the assumption of \(f\) being a module homomorphism, since

\begin{align*} f((m_i) \times (0) + (m_j) \times (0)) =& f(\iota(m_i)) + f(\iota(m_j)) \\ =& f_i(m_i) + f_j(m_j) \end{align*}

which then follows by induction