# Quotient Ring of Integers with Principal Ideal

## Theorem

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

Let $n \in \Z$.

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

The quotient ring $\struct {\Z, +, \times} / \ideal n$ is isomorphic to $\struct {\Z_n, +_n, \times_n}$, the ring of integers modulo $n$.

Note the special cases where $n = 0$ or $1$:

Quotient Ring of Integers and Zero
Quotient Ring of Integers and Principal Ideal from Unity