Definition:Integers Modulo m
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Definition
Let $m \in \Z$ be an integer.
The quotient set of congruence modulo $m$ is:
- $\displaystyle \Z_m = \frac {\Z} {\mathcal R_m} = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$
where:
- $\mathcal R_m$ is the equivalence relation defined as congruence modulo $m$
- $\left[\!\left[{x}\right]\!\right]_m$ is the residue class of $x$ modulo $m$.
Thus there are $m$ different residue classes modulo $m$.
From Congruence to an Integer less than Modulus, it follows that the set defined here is a complete repetition-free list of them.
This definition is a refinement of the concept of the set of all residue classes in the domain of real numbers.
This structure can also be rendered $\left({\N_m, +_m}\right)$, using $\N_m$ as defined in Subset of Natural Numbers.
Also see
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): Chapter $1 \ \S 1$: Example $4$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.5$
- Seth Warner: Modern Algebra (1965): $\S 7$: Example $7.1$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $2$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 18$
- Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.3$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 18$
