Last Updated：2022/12/24
A free abelian group of rank n is isomorphic to ℤ⊕ℤ⊕...⊕ℤ= bigoplus ⁿℤ, where the ring of integers ℤ occurs n times as the summand. The rank of a free abelian group is the cardinality of its basis. The basis of a free abelian group is a subset of it such that any element of it can be expressed as a finite linear combination of elements of such basis, with the coefficients being integers. (For an element a of a free abelian group, 1a = a, 2a = a + a, 3a = a + a + a, etc., and 0a = 0, (−1)a = −a, (−2)a = −a + −a, (−3)a = −a + −a + −a, etc.)
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A
free
abelian
group
of
rank
n
is
isomorphic
to
ℤ⊕ℤ⊕...⊕ℤ=
bigoplus
ⁿℤ,
where
the
ring
of
integers
ℤ
occurs
n
times
as
the
summand.
The
rank
of
a
free
abelian
group
is
the
cardinality
of
its
basis.
The
basis
of
a
free
abelian
group
is
a
subset
of
it
such
that
any
element
of
it
can
be
expressed
as
a
finite
linear
combination
of
elements
of
such
basis,
with
the
coefficients
being
integers.
(For
an
element
a
of
a
free
abelian
group,
1a
=
a,
2a
=
a
+
a,
3a
=
a
+
a
+
a,
etc.,
and
0a
=
0,
(−1)a
=
−a,
(−2)a
=
−a
+
−a,
(−3)a
=
−a
+
−a
+
−a,
etc.)
