Initial Object in Ring

1. Satz

Let \(\mathrm{CRing}\) be the category Ring, then the integers \(\mathbb{Z}\) are the initial object.

2. Beweis

Let \(R\) be a ring with \(1_R \in R\).

2.1. Existence

Let

\begin{align*} \varphi: \mathbb{Z} \rightarrow& R \\ 1 \mapsto& 1_R \\ n \mapsto& \sum_{i=1}^{n} (\mathrm{sgn}(n) 1) \end{align*}

2.2. Unique

Suppose \(\varphi, \psi: \mathbb{Z} \rightarrow R\) are ring homomorphism. Then by assumption \(\varphi(1) = \psi(1) = 1_R\) Hence for \(n \in \mathbb{Z}\) we conclude, that

\begin{align*} \varphi(n) =& \varphi(\sum_{i=1}^{n} \mathrm{sgn}(n) 1) \\ =& \sum_{i=1}^{n} \mathrm{sgn}(n) 1 \\ =& \psi(\sum_{i=1}^{n} \mathrm{sgn}(n) 1) \\ =& \psi(n) \end{align*}