Exercise 44, § 1

Let A be a ring. a) The set $\hat{U}(A)$ of elements of $U(A)$ with nilpotent coordinates, zero except for a finite number of them, is an ideal of the ring $U(A)$. For every $n \in J$, it is stable under $F_n$ and $V_n$. b) Let $a \in \hat{U}(A)$. Then the element $E(a)$ of $\Lambda(A)$ is a polynomial in T. Its value at $-1$ is an invertible element of A, which is also, for every $n \in J$, the value at $-1$ of the polynomial $E(V_n a)$. c) If $a \in U(A)$ and $b \in \hat{U}(A)$, we denote by $\langle a, b \rangle$ the value at $-1$ of the polynomial $E(ab)$. Thus one defines a $\mathbf{Z}$-bilinear mapping of $U(A) \times \hat{U}(A)$ into $A^*$, and one has