free abelian group

Noun

 * 1)  a free module over the ring of integers
 * A free abelian group of rank n is isomorphic to $$\mathbb{Z} \oplus \mathbb{Z} \oplus ... \oplus \mathbb{Z} = \bigoplus^n \mathbb{Z} $$, where the ring of integers $$\mathbb{Z}$$ 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, (&minus;1)a = &minus;a, (&minus;2)a = &minus;a + &minus;a, (&minus;3)a = &minus;a + &minus;a + &minus;a, etc.)

Hypernyms

 * free module