Addition and multiplication on $\mathbb{Z}/n$ 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 $\mathbb Z/n$ (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$) define

$[a]_n+[b]_n:=[a+b]_n,\qquad [a]_n[b]_n:=[ab]_n.$

These are binary operations in the sense of Binary operation on a set; associativity, commutativity, and a subset closed under the operation. Indeed, if $[a]_n=[a']_n$ and $[b]_n=[b']_n$, then $a\equiv a'\pmod n$ and $b\equiv b'\pmod n$, so

$a+b\equiv a'+b'\pmod n,\qquad ab\equiv a'b'\pmod n$

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 $\mathbb Z\to\mathbb Z/0$. 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 $\mathbb Z/1$ from being a field.