# Quotient Ring of Integers and Zero

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## Theorem

Let $\struct {\Z, +, \times}$ be the integral domain of integers.

Let $\ideal 0$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $0$.

The quotient ring $\struct {\Z / \ideal 0, +, \times}$ is isomorphic to $\struct {\Z, +, \times}$.

## Proof

\(\displaystyle \ideal 0\) | \(=\) | \(\displaystyle \set {\sum^n_{i \mathop = 1} r_i \times 0 \times s_i: n \in \N; r_i, s_i \in \Z}\) | Definition of Principal Ideal of Ring | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \set {\sum^n_{i \mathop = 1} 0: n \in \N}\) | $0$ is the zero under integer multiplication | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \set 0\) | Integer Addition Identity is Zero |