Axiomatic WikiWWikiDDefinitionsThe integers modulo n
Definition·D114
The set of residue classes of the integers under congruence modulo n.
For
, let
be the relation of congruence modulo $n$ on
(a set by Separation). The integers modulo
are
the quotient of
by
; write
for the class of
, its residue class.
In words
The integers modulo n are the quotient of the integers by congruence modulo n: the residue class of a collects every integer congruent to a into a single point of the quotient.
Never needed: F03 · F04 · F05 · F06 · F08 · F10 · F11 · F12 · F13 · A02 · A03 · A04 · A05 · A07 · A08 · A09 (computed from the citation graph, not asserted).
Remarks
By L73,
really is an equivalence relation, which is what makes this quotient meaningful; its classes are the cosets
. Boundary cases: for
, congruence is equality, so
is a faithful copy of
; for
, everything is congruent, so
is a single point. For
the quotient is finite with exactly
elements
(T60). Addition and multiplication descend representative-wise (D115, D116), making it the ring
(T59). The glyph
records that this is equally the quotient of the group
by the subgroup
(L72).
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