- #1
Bashyboy
- 1,421
- 5
Homework Statement
First Claim: If ##u \in R## is a unit, then it cannot be nilpotent
Second Claim: If ##u \in R## is nilpotent, then it cannot be a unit
Homework Equations
The Attempt at a Solution
I realize these are simple problems, but I have no one to verify my work and I want to be certain I am doing things correctly. Here is a proof of the first claim:
Suppose the contrary, that ##u## is a unit and also nilpotent. This implies there exists an element ##v## that acts as a multiplicative inverse and a natural number ##n## such that ##u^n = 0##. By the well ordering property, we can take ##n## to be the smallest natural number for which ##u^n=0##. Then
##u^n = 0##
##u u^{n-1} = 0##
##vu u^{n-1} = v0##
##u^{n-1} = 0##,
contradicting the minimality of ##n##.
Does this seem right? If I am not mistaken, then proof of the second claim is identical.