${\mathbb{Z}}_n$

Let $n\in \mathbb{Z}$. An equivalence relation, called congruence (http://planetmath.org/Congruent2), can be defined on $\mathbb{Z}$ by $a\equiv b\mathrm{mod}n$ iff $n$ divides $b-a$. Note first of all that $a\equiv b\mathrm{mod}n$ iff $a\equiv b\mathrm{mod}|n|$. Thus, without loss of generality, only nonnegative $n$ need be considered. Secondly, note that the case $n=0$ is not very interesting. If $a\equiv b\mathrm{mod}0$, 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 (http://planetmath.org/PrimeNumber) $p$, as ${\mathbb{Z}}_p$ is used to refer to the $p$-adic integers (http://planetmath.org/MathbbZ_p). 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 ${𝔽}_p$ when $n=p$ for some prime $p$ and ${\mathbb{Z}}_n$ otherwise. (The explanation of the usage of ${𝔽}_p$ will come later.) Still others, especially those who are unfamiliar with the , 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 (http://planetmath.org/EquivalenceClass), ) with the unique element $r$ in the class such that . 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 (http://planetmath.org/MultiplicativeIdentity) 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 ${𝔽}_p$ highlights the fact that the 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 (http://planetmath.org/Equivalent), yielding only one equivalence class.

The $n$ in both ${\mathbb{Z}}_n$ and $a\equiv b\mathrm{mod}n$ is called the modulus. Performing computations such as addition, subtraction, multiplication, and taking exponents (http://planetmath.org/Exponent2) in one of the rings ${\mathbb{Z}}_n$ is called modular arithmetic.