I8. Let (R, 1.) be a commutative rinK with identity and let N denote the set of nilpotent elements of R. Verify that a) the triple (N,+,) is an ideal of (R,+,). [Hint: If a - - 0, consider (a - b)*+) b) the quotient ring (R/N.+.) has no nonzero nilpotent elements.

I8. Let (R, 1.) be a commutative rinK with identity and let N denote the set of nilpotent elements of R. Verify that a) the triple (N,+,) is an ideal of (R,+,). [Hint: If a - - 0, consider (a - b)*+) b) the quotient ring (R/N.+.) has no nonzero nilpotent elements.

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 31E: Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set...

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I8. Let (R, 1.) be a commutative ring with identity and let N denote the set of
nilpotent elements of R. Verify that
a) the triple (N,+,) is an ideal of (R,+,). [lint: If a* - - 0, consider
(a – b)*+".)
b) the quotient ring (R/N, +, ) has no nonzero nilpotent elements.

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