Definition:Ring of Integers Modulo m

Definition

Let $m \in \Z: m \ge 2$.

Let $\Z_m$ be the set of integers modulo $m$.

Let $+_m$ and $\times_m$ denote addition modulo $m$ and multiplication modulo $m$ respectively.

The algebraic structure $\struct {\Z_m, +_m, \times_m}$ is the ring of integers modulo $m$.

Also denoted as

When the operations are understood to be $+_m$ and $\times_m$, it is usual to use just $\Z_m$ to denote the ring of integers modulo $m$.

The notation $\Z / m$ and $\Z / m \Z$ are also seen, deriving from Quotient Ring of Integers by Integer Multiples.

Cayley Tables for $\Z_2$

The Cayley tables for the Ring of Integers Modulo $2$ are as follows:

$\begin{array} {r|rr} \struct {\Z_2, +_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array} \qquad \begin{array}{r|rr} \struct {\Z_2, \times_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 0 2 \\ \eqclass 1 2 & \eqclass 0 2 & \eqclass 1 2 \\ \end{array}$

They can be presented more simply as:

$\begin{array}{r|rr} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{r|rr} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$

Also see

$\struct {\Z_m, +_m, \times_m}$ is a commutative ring with unity $\eqclass 1 m$
the zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.
• Results about the ring of integers modulo $m$ can be found here.

Notation

When $p$ is a prime number, the notation $\Z_p$ is used for the ring of integers modulo $p$.

The notation $\Z_p$ is also used for the $p$-adic integers.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the context of any page where $\Z_p$ appears will define what is referred to by $\Z_p$.