We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operati...

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer’s book [20]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930’s. In [17] and [18] the current authors investigated properties of the Kronecker functi...

We study the “local” behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the “global” problem of building a new semistar operation on a given integral domain, by “gluing” a given homogeneous family of semistar operations defined on a set of localizations. We apply these results for studying the local–global beh...

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type [⋆] on the polynomial ring D[X], such that D is a ⋆-quasi-Prüfer domain if and only if each upper to zero in D[X] is a quasi-[⋆]-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott [18, ...

Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequal...

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e, its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content cD(g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with cD(g) v = D. Using these facts, the...

let $d$ be an integral domain and $star$ a semistar operation stable and of finite type on it. we define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong s-domains. as an application, we give new characterizations of $star$-quasi-pr"{u}fer domains and um$t$ domains in terms of dimension ine...

If D is an integral domain with quotient field K, then let F̄(D) be the set of non-zero D-submodules of K, F(D) be the set of non-zero fractional ideals of D and f(D) be the set of non-zero finitely generated D-submodules of K. A semistar operation ? on D is called arithmetisch brauchbar (or a.b.) if, for every H ∈ f(D) and every H1, H2 ∈ F̄(D), (HH1) ? = (HH2) ? implies H 1 = H ? 2 , and ? is ca...

This paper studies the notion of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a ∗-maximal ideal and when a ∗-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩R 6= 0, for a given star operation of finite character ∗ on R[X]. We also answer negatively some questions raised by Anderson-Clarke by constr...