unity plus nilpotent is unit
If , then , which is a unit. Thus, we may assume that .
(Note that the summations include the term , which is why is excluded from this case.)
Note that there is a this proof and geometric series: The goal was to produce a multiplicative inverse of , and geometric series yields that
provided that the summation converges (http://planetmath.org/AbsoluteConvergence). Since is nilpotent, the summation has a finite number of nonzero terms and thus .
|Title||unity plus nilpotent is unit|
|Date of creation||2013-03-22 15:11:54|
|Last modified on||2013-03-22 15:11:54|
|Last modified by||Wkbj79 (1863)|