Definition:Modulo Multiplication/Definition 1
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Definition
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$:
- $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$
where $\eqclass x m$ is the residue class of $x$ modulo $m$.
The operation of multiplication modulo $m$ is defined on $\Z_m$ as:
- $\eqclass a m \times_m \eqclass b m = \eqclass {a b} m$
Also denoted as
Although the operation of multiplication modulo $m$ is denoted by the symbol $\times_m$, if there is no danger of confusion, the conventional multiplication symbols $\times, \cdot$ etc. are often used instead.
The notation for multiplication of two integers modulo $m$ is not usually $\eqclass a m \times_m \eqclass b m$.
What is more normally seen is $a b \pmod m$.
Also see
- Results about modulo multiplication can be found here.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes
- 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: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 18 \alpha$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 18 \ (2)$: Congruence classes
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.3$: Congruences
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.30$