Star-Invertibility and tt-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.Comment: 16 page

Suprema in spectral spaces and the constructible closure

Given an arbitrary spectral space $X$, we endow it with its specialization order $\leq$ and we study the interplay between suprema of subsets of $(X,\leq)$ and the constructible topology. More precisely, we investigate about when the supremum of a set $Y\subseteq X$ exists and belongs to the constructible closure of $Y$. We apply such results to algebraic lattices of sets and to closure operations on them, proving density properties of some distinguished spaces of rings and ideals. Furthermore, we provide topological characterizations of some class of domains in terms of topological properties of their ideals

Some topological considerations on semistar operations

We consider properties and applications of a new topology, called \emph{the Zariski topology}, on the space $\mathrm{SStar}(A)$ of all the semistar operations on an integral domain $A$. We prove that the set of all overrings of $A$, endowed with the classical Zariski topology, is homeomorphic to a subspace of $\mathrm{SStar}(A)$. The topology on $\mathrm{SStar}(A)$ provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace $\mathrm{SStar}_f(A)$ of all the semistar operations of finite type on $A$ is a spectral space

Invertibility of ideals in Pr\ufcfer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush and Zhang in [9], we investigate about connections between faithfully atness and invertibility for ideals in rings with zero divisors