Addition and multiplication on $\mathbb{Z}\mathrm{/}n$ by $[a{]}_n+[b{]}_n=[a+b{]}_n$ and $[a{]}_n[b{]}_n=[ab{]}_n$

Let $n\in \mathbb{N}$, including $n=0$ and $n=1$. For classes in $\mathbb{Z}\mathrm{/}n$ (The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$) define

$[a{]}_n+[b{]}_n:=[a+b{]}_n,[a{]}_n[b{]}_n:=[ab{]}_n\mathrm{.}$

These are binary operations in the sense of Binary operation on a set; associativity, commutativity, and a subset closed under the operation. Indeed, if $[a{]}_n=[a^{\mathrm{\prime }}{]}_n$ and $[b{]}_n=[b^{\mathrm{\prime }}{]}_n$, then $a\equiv a^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}n)$ and $b\equiv b^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}n)$, so

$a+b\equiv a^{\mathrm{\prime }}+b^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}n),ab\equiv a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}n)$

by Congruent integers may be added, subtracted and multiplied: representative changes preserve both arithmetic operations. Thus the two displayed classes do not depend on the representatives chosen.

For $n=0$, every class is a singleton and these operations reproduce integer addition and multiplication under the bijection $\mathbb{Z}\to \mathbb{Z}\mathrm{/}0$. For $n=1$, the quotient has one element, so $[0{]}_1=[1{]}_1$ and both operations are the unique binary operation on a one-element set.

• The representative-independence argument is part of the definition. Without it the displayed formulas would be rules on integers, not operations on classes.
• No field assertion is made here. In particular, the coincidence $[0{]}_1=[1{]}_1$ will exclude $\mathbb{Z}\mathrm{/}1$ from being a field.