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 $\left({\Z_m, +_m, \times_m}\right)$ 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:


 * $\left({\Z_m, +_m, \times_m}\right)$ is a commutative ring with unity $\left[\!\left[{1}\right]\!\right]_m$
 * the zero of $\left({\Z_m, +_m, \times_m}\right)$ is $\left[\!\left[{0}\right]\!\right]_m$.