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.
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.