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Question Number 12843 by tawa last updated on 04/May/17

Let R be a cummutative ring with 1. and a,b  member of R. Suppose a is  invertible and b is nilpotent. Show that a + b is invertible.

$$\mathrm{Let}\:\mathrm{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{cummutative}\:\mathrm{ring}\:\mathrm{with}\:\mathrm{1}.\:\mathrm{and}\:\mathrm{a},\mathrm{b}\:\:\mathrm{member}\:\mathrm{of}\:\mathrm{R}.\:\mathrm{Suppose}\:\mathrm{a}\:\mathrm{is} \\ $$$$\mathrm{invertible}\:\mathrm{and}\:\mathrm{b}\:\mathrm{is}\:\mathrm{nilpotent}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{a}\:+\:\mathrm{b}\:\mathrm{is}\:\mathrm{invertible}. \\ $$

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