Math 514 homework #4 Due date: 19950906 D. J. Bernstein 19950830 A module is a commutative group. Let F be a field. Any nonempty subset of F closed under subtraction is a module. If M ⊆ F is a module and z is an element of F then the scalar product zM = {zx : x ∈ M} is also a module. Let M and N be modules in F. Then the sum M + N = {x + y : x ∈ M, y ∈ N} and ratio M ÷ N = {z ∈ F : zN ⊆ M} are modules. The product MN is the smallest module containing {xy : x ∈ M, y ∈ N}. In the following exercises M, N, and T are modules in some field F. Exercise 9. Prove that T (M + N) = T M + T N. Exercise 10. Prove that M(NT) = (MN)T. (Hint: Show that M(NT) is the smallest module containing {xyz : x ∈ M, y ∈ N, z ∈ T }.) Exercise 11. Prove that (M ÷ N) ÷ T = M ÷ NT. Exercise 12. Put α = (1 + \sqrt{−35})/2, so that α^{2} − α + 9 = 0. Consider the ring R = Z[α]. Prove that (3R + αR)^{2} = αR.