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...

Need help with Artinian rings for ABstract ALgebra please.

Extracted 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.]