Exercise 9, § 6

a) An element $x$ (resp. an ideal $a$) of a topological ring $A$ is said to be topologically nilpotent if the sequence $(x^n)_{n \geq 1}$ converges to $0$ [resp. if the filter base formed by the $a^n$ ($n \geq 1$) converges to $0$]. Every element of a topologically nilpotent ideal is topologically nilpotent.

b) Suppose that $A$ has an identity element and that $A^*$ is open in $A$. Show that if $x$ is any topologically nilpotent element of $A$, then $1 - x$ is a unit (observe that $1 - x^n$ is a unit for sufficiently large $n$). Hence show that if all the elements of a (left or right) ideal $a$ are topologically nilpotent, then $a$ is contained in the radical of $A$.

c) Suppose that $A$ is complete and has an identity element, and that there exists a fundamental system of neighbourhoods of $0$ which are subgroups of the additive group of $A$. Show that if $x$ is topologically nilpotent, then $1 - x$ is a unit in $A$.

d) Under the hypotheses of c), show that if $y \in A$ is topologically nilpotent, then the equation $x^2 + x = y$ has a root $x \in A$, and that $x$ is topologically nilpotent.