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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