QuotientRing

The ring $\mathbb{Z}/m\mathbb{Z}$: compute-engine's QuotientRing, made the finite collection of its $m$ ResidueClass classes.

QuotientRing(Integers, m)$\mathbb{Z}/m\mathbb{Z}$ — Sage's Zmod(m)

Typed signatures

Type	Package	Applies
(set<any>, any) -> set	compute-engine	always

Domain: Modular arithmetic

Details

• A collection: it counts, enumerates and answers membership, so Count, ListFrom and Element work on it directly
• Membership is by modulus — ResidueClass(3, 7) is not in $\mathbb{Z}/5\mathbb{Z}$
• Its elements are residue classes, so it is typed set<value>, not compute-engine's set<integer>
• Another base ring, or a symbolic $m$, stays inert, as compute-engine has it
• Written $\mathbb{Z}/m\mathbb{Z}$, which reads back as itself; IntegerModRing(m) is its old spelling

Examples

3 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{ListFrom}(ℤ/3ℤ)$Out$\bigl\lbrack\overline{0}_{3}, \overline{1}_{3}, \overline{2}_{3}\bigr\rbrack$

live ↗

In$\vert ℤ/12ℤ\vert$Out$12$

Scope #

$\mathbb{Z}/12\mathbb{Z}$

live ↗

In$ℤ/12ℤ$Out$ℤ/12ℤ$

Properties #

live ↗

In$\overline{3}_{5}∈ℤ/5ℤ$Out$\mathrm{True}$

About

Examples	7

See also: ResidueClass, IntegerModRing