Constructing units from nilpotent elements in a commutative ring

Let R be a commutative ring with 1 \neq 0. Prove that if a \in R is nilpotent, then 1-ab is a unit for all b \in R.


By this previous exercise, \mathfrak{N}(R) is an ideal of R. Thus for all b \in R, -ab is nilpotent. By this previous exercise, 1 - ab is a unit in R.

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