A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R, i.e. whenever I, 2 1, 2 13 2 .… is a decreasing chain of ideals of R, then there is a positive integer m such that I = Im for all k > m. ... %3D Let R be a commutative Artinian ring with identity 1 + 0. (a) Show that every nonzero element of R is either a unit or a zero divisor. (b) Show that every prime ideal of R is maximal. [Hint: show that R/P is Artinian.]
A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R, i.e. whenever I, 2 1, 2 13 2 .… is a decreasing chain of ideals of R, then there is a positive integer m such that I = Im for all k > m. ... %3D Let R be a commutative Artinian ring with identity 1 + 0. (a) Show that every nonzero element of R is either a unit or a zero divisor. (b) Show that every prime ideal of R is maximal. [Hint: show that R/P is Artinian.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with Artinian rings for ABstract ALgebra please.
![5) A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R, i.e.
whenever I, 2 l, 2 I3 2 .…· is a decreasing chain of ideals of R, then there is a positive integer m
such that I = Im for all k > m.
Let R be a commutative Artinian ring with identity 1 + 0.
(a) Show that every nonzero element of R is either a unit or a zero divisor.
(b) Show that every prime ideal of R is maximal. [Hint: show that R/P is Artinian.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F422648a4-0aba-459c-969d-63777652c5b7%2Fzszxy6q_processed.png&w=3840&q=75)
Transcribed Image Text:5) A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R, i.e.
whenever I, 2 l, 2 I3 2 .…· is a decreasing chain of ideals of R, then there is a positive integer m
such that I = Im for all k > m.
Let R be a commutative Artinian ring with identity 1 + 0.
(a) Show that every nonzero element of R is either a unit or a zero divisor.
(b) Show that every prime ideal of R is maximal. [Hint: show that R/P is Artinian.]
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