Question

modules which are neither Noetherian nor Artinian.is it true of false.if it is one them then give proof

Answer 1

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.