free abelian group

== English == === Noun === free abelian group (plural free abelian groups) (algebra) A 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.) ==== Hypernyms ==== free module