zero object in RMod
1. Proposition
Let \(R\) be a ring and \(\mathrm{RMod}\) the category RMod Then the zero module is the zero object
2. Proof
Let \(M\) be an \(R\)-module
2.1. terminal object
Then the map
\begin{align*} 0: M \rightarrow& 0 \\ m \mapsto& 0 \\ \end{align*}is a welldefined module-homomorphism
Furthermore, it is the only set theoretic map, hence unique
2.2. initial object
Then the map
\begin{align*} 0: 0 \rightarrow& M \\ 0 \mapsto& 0 \\ \end{align*}is a welldefined
furthermore, by
\begin{align*} \varphi(0) = 0 \end{align*}this map is unique