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 $1$:

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


Proof


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Sources

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