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:

Also see

 * Ring of Integers Modulo m is Ring, where it is shown that:


 * $\struct {\Z_m, +_m, \times_m}$ is a commutative ring with unity $\eqclass 1 m$
 * the zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.

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 the context of any page where $\Z_p$ appears will define what is referred to by $\Z_p$.