Sufficient Condition to Resolve Costa's First Conjecture

Abstract

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d − 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d − 1)-ring (resp., neither a (5, d − 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.

Key Words:

• Costa's first conjecture
• (n, d)-Ring
• Trivial extension

2000 Mathematics Subject Classification:

• 13D02
• 13D05
• 13D07
• 13B02

ACKNOWLEDGMENTS

I would like to express my sincere thanks for the referee for his/her helpful suggestions and comments, which have greatly improved this article.

Notes

Communicated by R. Wiegand.

Dedicated to my advisor Salah Eddine Kabbaj.