Answer to Question #107486 in Abstract Algebra for Irshad

Question #107486
modules which are neither Noetherian nor Artinian.is it true of false.if it is one them then give proof
1
Expert's answer
2020-04-03T10:30:59-0400

Consider the "\\R" -module "M=\\R[x_1,x_2,x_3,...........]"

Claim: "M" is neither Noetherian nor Artinian.

Consider the descending chain of submodule

"(x_1)\\supe({x_1}^2)\\supe({x_1}^3)\\supe....................."

Which is an infinite descending chain of submodule.

Hence,"M" is not a Artinian module.

Again ,Consider the ascending chain of submodule

"(x_1)\\sube(x_1,x_2)\\sube(x_1,x_2,x_3)\\sube............"

Which is a infinite ascending chain of submodule.

Hence ,"M" is not a Noetherian module.

Hence, "M" is neither Noetherian nor Artinian module.



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS