Proposition 1

— Let A be a commutative ring and x an element of A. For x to be nilpotent it is necessary and sufficient that $1 - xT$ be invertible in the ring $A[T]$.

We note that $A[T]$ is a subring of the formal power series ring $A[[T]]$ and that $1 - xT$ has in $A[[T]]$ the inverse $\sum_{n=0}^{\infty} x^n T^n$ (IV, p. 30, Prop. 5). For $1 - xT$ to be invertible in $A[T]$ it is necessary and sufficient for $\sum_{n=0}^{\infty} x^n T^n$ to be a polynomial, that is, for x to be nilpotent.