Polynomial extensions of semistar operations
Let \(D\) be an integral domain. The paper under review is devoted to the following problem: how to extend ``in a canonical way'' an arbitrary semistar operation of \(D\) to the polynomial ring \(D[X]\). The first attempts were done by \textit{G. Picozza} [Houston J. Math. 33, No. 2, 415--432 (2007; Zbl 1114.13006)] and then by the first two authors of this paper [J. Algebra 318, No. 1, 484--493 (2007; Zbl 1136.13011) and Commun. Algebra 39, No. 6, 1956--1978 (2011; Zbl 1234.13011)]. Their study is focused on the stable semistar operations of finite type. In this paper, the authors provide a complete solution to this problem in the most general setting. As an application, they show that, in the particular cases of stable semistar operations of finite type or semistar operations defined by families of over-rings.
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