4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2. Show 1 - s is a unit of S. Hint: Expand the product (1s) (1+s+...+ sn−1). (b) Use (a) above to find a commuative ring R with 1 and a non-constant polynomial ƒ € R[x] such that f is a unit of R[x].

4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2. Show 1 - s is a unit of S. Hint: Expand the product (1s) (1+s+...+ sn−1). (b) Use (a) above to find a commuative ring R with 1 and a non-constant polynomial ƒ € R[x] such that f is a unit of R[x].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 33E: 33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all...

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