Semistar invertibility on integral domains

Radical semistar operations

We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that $D$ is rad-colon coherent). When $D$ is a Prüfer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.

Minimal Prime Ideals and Semistar Operations

Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal, then the set $\Min(I)$ of minimal prime ideals over $I$ is finite.

On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations

Let $D$ be an integral domain with quotient field $K$. The $b$-operation that associates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid V valuation overring of D\}$, is a semistar operation that plays an important role in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$, $I^b$ coincides with its integral closure). In a first part of the paper, we study the integral domains that are $b$-Noetherian (i.e., such that, for each nonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$ of $D$). For instance, we prove that a $b$-Noetherian …

Semistar dimension of polynomial rings and Prüfer-like domains

Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper 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-Prüfer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domains). We also extend Arnold's formula to the setting of semistar operations.

Semistar-Krull and Valuative Dimension of Integral Domains

Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\star$-$\dim(D)$, then $n+1\leq \star[X]\text{-}\dim(D[X])\leq 2n+1$. We also establish that if $D$ is a $\star$-Noetherian domain or is a Prüfer $\star$-multiplication domain, then $\star[X]\text{-}\dim(D[X])=\star\text{-}\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\star$-$\dim_v(D)$, to be the maximal …

Descending chains of semistar operations

A class of integer-valued functions defined on the set of ideals of an integral domain $R$ is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the $e$-operation. We use this class of functions to recover familiar semistar operations such as the $w$-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.

Nagata Rings, Kronecker Function Rings and Related Semistar Operations

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 \cite{G}) 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 \cite{FL1} and \cite{FL2} the current authors investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain $D$. In this paper we extend that study and also generalize Kang's notion of a star Nagata ring \cite{Kang:19…

分次整环上保持齐次性的半星运算空间

The space of homogeneous preserving semistar operations on graded domains

本文研究分次整环上保持齐次性的半星运算空间，证明 $\star_a$ 保持齐次性，并证明齐次 Kronecker 函数环的赋值超环集合与分次赋值超环集合同胚，以及有限型保持齐次性的半星运算集合构成谱空间。

Near prequantales and applications to star and semistar operations and ideal and module systems

We show that a generalization of the theory of quantales and prequantales provides a noncommutative and nonassociative abstract ideal theoretic setting for the theories of star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results in the theory of quantales and prequantales. Applications include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of simple commutative quantales and simple multiplicative lattices; a construction of the largest finitary …

Uppers to zero in polynomial rings and Prüfer-like domains

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 $g \in D[X]$ with content $\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \in D[X]$ with $\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Prüfer domain can be naturally extended to the sem…

Star-Invertibility and $t$-finite character in Integral Domains

Let $A$ be an integral domain. We study new conditions on families of integral ideals of $A$ in order to get that $A$ is of $t$-finite character (i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal ideals). We also investigate problems connected with the local invertibility of ideals.

Non-compact subsets of the Zariski space of an integral domain

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.

Characterizations of graded Prüfer $\star$-multiplication domains

Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain graded by an arbitrary grading torsionless monoid $Γ$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$ with respect to $\star$. We say that $R$ is a graded Prüfer $\star$-multiplication domain if each nonzero finitely generated homogeneous ideal of $R$ is $\star_f$-invertible. Using $\star$-Nagata and Kronecker function rings, we give several different equivalent conditions for $R$ to be a graded Prüfer $\star$-multipli…

Some remarks on Prüfer $\star $--multiplication domains and class groups

Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\boldsymbol{\mathcal{T}}:=\{T_λ\mid λ \in Λ\}$ be a defining family of quotient rings of $D$ and suppose that $\ast $ is a finite type star operation on $D$ induced by $\boldsymbol{\mathcal{T}}$. We show that $D$ is a P$ \ast $MD (resp., P$v$MD) if and only if $(\co_D(fg))^{\ast}=(\co_D(f)\co_D(g))^{\ast}$ (resp., $(\co_D(fg))^{w}=(\co_D(f)\co_D(g))^{w}$) for all $0 \ne f,g \in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for re…

Characterizations of graded Prüfer $\star$-multiplication domains, II

Let $R=\bigoplus_{α\inΓ}R_α$ be a graded integral domain and $\star$ be a semistar operation on $R$. For $a\in R$, denote by $C(a)$ the ideal of $R$ generated by homogeneous components of $a$ and for$f=f_0+f_1X+\cdots+f_nX^n\in R[X]$, let $\A_f:=\sum_{i=0}^nC(f_i)$. Let $N(\star):=\{f\in R[X]\mid f\neq0\text{and}\A_f^{\star}=R^{\star}\}$. In this paper we study relationships between ideal theoretic properties of $\NA(R,\star):=R[X]_{N(\star)}$ and the homogeneous ideal theoretic properties of $R$. For example we show that $R$ is a graded Prüfer-$\star$-multiplication domain if and only if $\NA…