On Serial Noetherian Rings

Abstract

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.

Key Words:

• Artinian module
• Hereditary ring
• Noetherian module
• Serial modules and rings
• Uniserial

AMS (2000) Mathematics Subject Classification:

• 16D50
• 16D70
• 16D80

ACKNOWLEDGMENTS

The authors would like to thank Professor R. Wisbauer for useful remarks and Professor Ch. Hajarnavis for many helpful comments, especially for the proof of Lemma 2.

Notes

Communicated by R. Wisbauer.