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

Abstract

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 fieldIn this paper, 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.

When the Nagata Ring D(x) Is a Sharp Domain

“…The following statements are equivalent for an integral domain D.Let be a star operation on D[X], and let I * = (ID[X]) ∩ K for all I ∈ F (D). Then it is easy to see that * is a star operation on D (cf [6,. Lemma 5]).…”

Polynomial extensions of semistar operations

“…In this paper, we provide a complete solution to this problem in the most general setting. As an application, we show that, in the particular cases of stable semistar operations of finite type or semistar operations defined by families of overrings, we reobtain the main results given in [1] and [2]. Finally, we investigate the behavior of the polynomial extensions of some operations among the most important and classical ones such as the d D , v D , t D , w D and b D operations.…”

“…The first attempts to extend a semistar operation of D to a semistar operation of D[X] were done by G. Picozza [15] and then by the first two authors of this paper [1,2]. Their study is focused on the stable semistar operations of finite type.…”