# Quotient Ring of Integers with Principal Ideal

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## 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 $1$:

## Proof

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Example $39$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 60$. Factor rings: Illustration