Exercise 2, § 4

Let K be a commutative field, E, F two fields whose centre contains

K, M a left vector space over E and N a left vector space over F; let G = E ⊗_K F.

(a) Let x ≠ 0 be an element of M and y ≠ 0 an element of N. Show that x ⊗ y is free in the G-module M ⊗_K N (use Exercise 1(d) and the transitivity of End_E(M) (resp. End_F(N)) in the set of elements of M (resp. N) distinct from 0).

(b) Show that M ⊗_K N is a free G-module. (If (m_α) is a basis of M over E and (n_β) a basis of N over F, show that the sum of the G-modules G(m_α ⊗ n_β) is direct, using a method analogous to that of (a).)