Every torsion-free divisible abelian group D is a direct sum of copies of the rationals Q. if O ≠ n e Z and α ϵ D, then there exists a unique b ϵ D such that nb = a. Denote h by (1/n)a. For m, n ϵ Z (n ≠ 0), define (m/n)a = m(1/n)a. Then D is a vector space over Q. Use Theorem 2.4.

Data from Theorem 2.4

Every vector space V over a division ring D has a basis and is therefore  a free D-module. More generally every linearly independent subset of V is contained in a basis of V.  The converse of Theorem 2.4 is also true, namely, if every unitary module over a  ring D with identity is free, then D is a division ring (Exercise 3.14).

Data from Exercise 3.14

Let R be a commutative ring with identity and prime characteristic p. The map  R →R given by r |→ r^P is a homomorphism of rings called the Frobenius homomorphism [see Exercise 11 ].

Data from Exercise 11

(The Freshman's Dream1). Let R be a commutative ring with identity of prime characteristic p. If a,b ϵ R, then for all integers n > 0 [see Theorem 1.6 and Exercise 10; note that b = -b if p = 2].

Data from Theorem 1.6Data from Exercise 10