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
- 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 $\eqclass 0 m$.
- 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 $\Z_p$.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Example $21.1$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $2$
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences