1.NounA free module over the ring of integers."A free abelian group of rank n is isomorphic to
Z
⊕
Z
⊕
.
.
.
⊕
Z
=
⨁
n
Z
{\displaystyle \mathbb {Z} \oplus \mathbb {Z} \oplus ...\oplus \mathbb {Z} =\bigoplus ^{n}\mathbb {Z} }
, where the ring of integers
Z
{\displaystyle \mathbb {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.)"