unity plus nilpotent is unit


If x is a nilpotent elementMathworldPlanetmath of a ring with unity 1 (which may be 0), then the sum 1+x is a unit of the ring.


If x=0, then 1+x=1, which is a unit. Thus, we may assume that x0.

Since x is nilpotent, there is a positive integer n such that xn=0. We multiply 1+x by another ring element:

(1+x)j=0n-1(-1)jxj = j=0n-1(-1)jxj+k=0n-1(-1)kxk+1
= j=0n-1(-1)jxj-k=1n(-1)kxk
= 1+j=1n-1(-1)jxj-k=1n-1(-1)kxk-(-1)nxn
= 1+0+0
= 1

(Note that the summations include the term  (-1)0x0, which is why x=0 is excluded from this case.)

The reversed multiplicationPlanetmathPlanetmath gives the same result. Therefore, 1+x has a multiplicative inverse and thus is a unit. ∎

Note that there is a this proof and geometric series: The goal was to produce a multiplicative inverse of 1+x, and geometric series yields that


provided that the summation converges (http://planetmath.org/AbsoluteConvergence). Since x is nilpotent, the summation has a finite number of nonzero terms and thus .

Title unity plus nilpotent is unit
Canonical name UnityPlusNilpotentIsUnit
Date of creation 2013-03-22 15:11:54
Last modified on 2013-03-22 15:11:54
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 21
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 13A10
Classification msc 16U60
Related topic DivisibilityInRings