An Overring-Theoretic Approach to Polynomial Extensions of Star and Semistar Operations

Abstract

Call a semistar operation ⊛ on the polynomial domain D[X] an extension (respectively, a strict extension) of a semistar operation ☆ defined on an integral domain D, with quotient field K, if E ^☆ = (E[X])^⊛ ∩ K (respectively, E ^☆[X] = (E[X])^⊛) for all nonzero D-submodules E of K. In this article, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on D[X] that are “canonical” extensions (or, “canonical” strict extensions) of semistar operations on D.

Key Words:

• Kronecker function ring
• Localizing system of ideals
• Nagata ring
• Polynomial domain
• Star and semistar operation
• v–
• t–
• w–
• b–operation

2000 Mathematics Subject Classification:

• 13B25
• 13A15
• 13G05
• 13B22

ACKNOWLEDGMENTS

We thank Muhammad Zafrullah for the helpful suggestions received during the preparation of the present article and the referee for providing constructive comments.

The second author was partially supported by a MIUR-PRIN grant no. 2008WYH9NY.

Notes

Communicated by T. Albu.