Let $n \in \mathbb{Z}$ . An equivalence relation, called congruence, can be defined on $\mathbb{Z}$ by $a \equiv b \operatorname{mod} n$ iff $n$ divides $b-a$ . Note first of all that $a \equiv b \operatorname{mod} n$ iff $a \equiv b \operatorname{mod} |n|$ . Thus, without loss of generality, only nonnegative $n$ need be considered. Secondly, note that the case $n=0$ is not very interesting. If $a \equiv b \operatorname{mod} 0$ , then $0$ divides $b-a$ , which occurs exactly when $a=b$ . In this case, the set of all equivalence classes can be identified with $\mathbb{Z}$ . Thus, only positive $n$ need be considered. The set of all equivalence classes of $\mathbb{Z}$ under the given equivalence relation is called ${\mathbb{Z}}_n$ .

Some mathematicians consider the notation ${\mathbb{Z}}_n$ to be archaic and somewhat confusing. This matter of notation is most considerable when $n=p$ for some prime $p$ , as ${\mathbb{Z}}_p$ is used to refer to the $p$ -adic integers. To avoid this confusion, some mathematicians use the notation $\mathbb{Z}/n\mathbb{Z}$ instead of ${\mathbb{Z}}_n$ . On the other hand, the notation ${\mathbb{Z}}_n$ should not cause confusion when $n$ is not prime, and is an intuitive shorthand way to write $\mathbb{Z}/n\mathbb{Z}$ . Thus, others use ${\mathbb{F}}_p$ when $n=p$ for some prime $p$ and ${\mathbb{Z}}_n$ otherwise. (The explanation of the usage of $\mathbb{F}_p$ will come later.) Still others, especially those who are unfamiliar with the $p$ -adic integers, use the notation ${\mathbb{Z}}_n$ exclusively. (In this entry, the notation ${\mathbb{Z}}_n$ is used exclusively, though it is highly recommended to use another notation when $n=p$ for some prime $p$ .)

One usually identifies an element of ${\mathbb{Z}}_n$ (which is technically a class, not a number) with the unique element in the class $r$ such that $0 \le r < n$ . One can use the division algorithm to establish that, for each class, an $r$ as described exists uniquely. (The set of all $r$ 's as described is an example of a residue system.) Thus, the sets ${\mathbb{Z}}_n$ are finite with exactly $n$ elements. Addition and multiplication operations can also be defined on ${\mathbb{Z}}_n$ in a natural way that corresponds to the operations on $\mathbb{Z}$ . Under these operations, ${\mathbb{Z}}_n$ is a commutative ring with identity as well as a cyclic ring with behavior $1$ . When $n=p$ for some prime $p$ , ${\mathbb{Z}}_n$ is a field. In this case, the notation ${\mathbb{F}}_p$ highlights the fact that the structure is a field. When $n$ is composite, ${\mathbb{Z}}_n$ has zero divisors and thus is neither a field nor an integral domain. Also note that ${\mathbb{Z}}_1$ is a zero ring, since all integers are equivalent, yielding only one equivalence class.

The $n$ in both ${\mathbb{Z}}_n$ and $a \equiv b \operatorname{mod} n$ is called the modulus. Performing computations such as addition, subtraction, multiplication, and taking exponents in one of the rings ${\mathbb{Z}}_n$ is called modular arithmetic.