Question

An element a of a ring r is called nilpotent if an = 0 for some n n. Show that ifu is a unit and a is nilpotent in r then u + a is a unit.

Answer 1

Answer:

As u is a unit, it has an inverse v such that uv=1.

As a is nilpotent, there is a positive integer n such that aⁿ=0.

We must show that (u+a)w = 1 for some w.

Let z = u^(n-1) - u^(n-2) a + u^(n-3) a^2 - ... + (-1)^(n-1) (a)^(n-1).

Then:

(u+a)z = uⁿ + (-1)^(n-1) aⁿ = uⁿ,  since aⁿ = 0.

Multiplying both sides by vⁿ gives

(u+a)zvⁿ = uⁿvⁿ = (uv)ⁿ = 1      ( assuming the ring is commutative! )

Thus u+a is a unit with inverse given by w = zvⁿ.