On polynomial ascent and descent semistar operations on an integral domain

In 1994, A. Okabe and R. Matsuda introduced the notion of a semistar operation in [OM] as a generalization of the notion of a star operation which was introduced in 1936 by W. Krull and was developed in [G] by R. Gilmer. In 2000, M. Fontana and J.A. Huckaba investigated the relation between semistar operations and localizing systems and they associated the semistar operation ∗F for each localizing system F on D and the localizing system F∗ for each semistar operation ∗ on D. Using these correspondences, they established a very natural bridge between semistar operations and localizing systems which has been proven to be a very important and essential tool in the study of semistar operation theory. Let D be an integral domain with quotient field K and let D[X] be the ring of polynomials over D in indeterminate X. We shall denote the set of all semistar operations on D (resp. D[X]) by SS(D) (resp. SS(D[X])) as in [O5]. We have much interest in considering the relation between SS(D) and SS(D[X]). First, in [OM], a correspondence ∗ 7→ ∗′ from SS(D[X]) into SS(D) was given by setting E∗ ′ = (ED[X])∗ ⋂ K for each nonzero D-submodule E of K. In this paper, this semistar operation ∗′ is called the polynomial descent semistar operation associated to ∗ and is denoted by ∗. Next, in [P3], G. Picozza defined a reverse correspondence ∗ 7→ ∗′ from SS(D) into SS(D[X]) by setting ∗′ = ∗F∗[X] for each ∗ ∈ SS(D). In this paper, this semistar operation ∗′ is called the polynomial ascent semistar operation associated to ∗ and is denoted by ∗. Thus we have two correspondences between SS(D) and SS(D[X]). The purpose of this paper is to investigate the relation between SS(D) and SS(D[X]) using these two semistar operations ∗ and ∗. In Section 1, we first recall some well-known results on semistar operations and localizing systems on an integral domain D which will be used in sequel and we shall show some new results concerning semistar operations [∗] and ∗a which were introduced in [FL1]. In Section 2, we shall prove some important properties of semistar operations ∗ and ∗. In Theorem 27, we show that (∗) = ∗̄ for each semistar operation ∗ on D