Exercise 1, § 2

Let E be an A-algebra with a basis of two elements $e_1, e_2$ and a unit element $e = \lambda e_1 + \mu e_2$.

(a) Show that the ideal $a$ of A generated by $\lambda$ and $\mu$ is the whole ring A (in the contrary case, note that $E/aE = E \otimes_A (A/a) = \{0\}$ would hold contrary to the fact that $A/a \neq \{0\}$ and E is a free A-module).

(b) If $\alpha, \beta$ are two elements of A such that $\alpha \lambda + \beta \mu = 1$, show that the elements $e$ and $u = \beta e_1 - \alpha e_2$ also form a basis of E and deduce that E is a quadratic A-algebra.