Home

qZ

qZ denotes the set of all integer multiples of q, where q is an integer. Formally, qZ = { qk : k ∈ Z }. For q ≠ 0, this is an infinite cyclic subgroup of the additive group of integers and, in ring-theory terms, a principal ideal of the ring Z generated by q. If q = 0, then qZ = {0}.

In more detail, qZ is closed under addition and subtraction and is the kernel of the natural

Examples help illustrate the concept. If q = 3, then 3Z = {..., -6, -3, 0, 3, 6, ...}, the

Related ideas include principal ideals in Z, cyclic groups, and modular arithmetic. The notation qZ is widely

reduction
map
from
Z
to
Z/qZ.
The
quotient
Z/qZ
is
a
finite
cyclic
group
(and
ring)
of
order
|q|
when
q
≠
0.
Consequently,
the
index
of
qZ
in
Z
is
|q|,
meaning
there
are
exactly
|q|
distinct
cosets
of
qZ
in
Z.
For
q
=
±1,
qZ
equals
Z
itself,
and
Z/qZ
is
the
trivial
group.
set
of
multiples
of
3.
The
quotient
Z/3Z
consists
of
the
residue
classes
[0],
[1],
and
[2],
forming
a
cyclic
group
of
order
3.
If
q
=
0,
0Z
=
{0}
and
Z/0Z
is
isomorphic
to
Z,
an
infinite
group.
used
to
discuss
divisibility,
residue
classes,
and
the
structure
of
subgroups
and
ideals
within
the
integers.