A free abelian group of rank n is isomorphic to Z ⊕ Z ⊕ . . . ⊕ Z = ⨁ n Z , 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, 1 a = a, 2 a = a + a, 3 a = a + a + a, etc., and 0 a = 0, (−1) a = − a, (−2) a = − a + − a, (−3) a = − a + − a + − a, etc.). Wiktionary