Conditions for a ring to be Noetherian or Artinian

ABSTRACT

Let M be a finitely generated module. It is shown that M is noetherian iff every finitely generated module in is a direct sum of a projective module and a module Q, where Q is either CS or noetherian.

Acknowledgments

We would like to thank Professor Sompong Dhompongsa for many useful discussions and helpful comments. Moreover the authors wish to thank the referee for his suggestion to include Lemma 2 and for his hard work on simplifying the proofs which brought our original paper to the present form.

The first author was supported by Thailand Research Fund under grant PDF/91/2544. The second author was supported by the Royal Golden Jubilee Program.