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.
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