Exercise 2, § 3

Consider the field $\mathbf{C}$ of complex numbers as a vector space over the field $\mathbf{R}$ of real numbers. (a) Show that the canonical mapping $\mathbf{C} \otimes_{\mathbf{R}} \mathbf{C} \to \mathbf{C} \otimes_{\mathbf{C}} \mathbf{C}$ is not injective. (b) Show that the two $\mathbf{C}$-module structures on $\mathbf{C} \otimes_{\mathbf{R}} \mathbf{C}$ arising from each of the factors are distinct.*