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Carmelo Antonio Finocchiaro

Publications and source records attributed to Carmelo Antonio Finocchiaro.

13 recordsLinked to original sources
arXiv2026

Spaces of subobjects as spectral spaces

We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $λ$-generated object and the order-theoretic notion of a $λ$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.

math.CT↗
arXiv2026

Digroups, their canonical pretorsion theory, and diheaps

In the category of digroups there is a natural pretorsion theory in which the torsion-free digroups are all groups, and torsion digroups form a category isomorphic to the category of non-empty sets. It is also possible to extend the theory of heaps from groups to digroups. The corresponding notion is that of a diheap.

math.GR↗
arXiv2026

A pretorsion theory for right groups

Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.

math.CT↗
arXiv2024

Some remarks on Prüfer rings with zero-divisors

Let $A$ be the fiber product $R\times_TB$, where $B\to T$ is a surjective ring homomorphism with regular kernel and $R\subseteq T$ is a ring extension where $T$ is an overring of $R$. In this paper we provide a characterization of when $A$ has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings.

math.AC↗
arXiv2024

On bi-amalgamated constructions

Let $f:A\longrightarrow B, g:A\longrightarrow C$ be ring homomorphisms and let $\mathfrak{b}$ (resp., $\mathfrak{c}$) be an ideal of $B$ (resp., $C$) satisfying $f^{-1}(\mathfrak{b})=g^{-1}(\mathfrak{c})$. Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring $$A\bowtie^{f,g}(\mathfrak{b},\mathfrak{c}) :=\{(f(a)+b, g(a)+c)\mid a\in A, b\in\mathfrak{b}, c\in \mathfrak{c} \}$$ of $B\times C$, called the bi-amalgamation of $A$ with $(B,C)$ along $(\mathfrak{b}, \mathfrak{c})$, with respect to $(f,g)$. This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D'Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante, by providing a deeper insigt on the ideal-theoretic structure of bi-amalgamations.

math.AC↗
arXiv2022

Multiplicative lattices: maximal implies prime and related questions

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of m-systems. The notion of m-system is also studied from the topological point of view.

math.RA↗
arXiv2021

Abstractly constructed prime spectra

The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum $\mathrm{Spec}(R)$ of a unital commutative ring $R$ is always a spectral (=coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of $R$ is played by elements of an abstract complete lattice $L$ equipped with binary multiplication with $xy\leqslant x\wedge y$ for all $x,y\in L$. In fact when no further conditions on $L$ are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of $L$; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately.

math.CT↗
arXiv2019

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.

math.GN↗
arXiv2016

Invertibility of ideals in Prufer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush-Zhang, we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.

math.AC↗
arXiv2016

Algebraic and topological properties of an amalgamated algebra along an ideal

Let $f:A \rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$, a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[\![X]\!]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamation and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen-Macaulay ring or a Gorenstein ring.

math.AC↗
arXiv2010

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.

math.AC↗
arXiv2009

Amalgamated algebras along an ideal

Let $f:A \to B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we initiate a systematic study of a new ring construction called the "amalgamation of $A$ with $B$ along $J$ with respect to $f$". This construction finds its roots in a paper by J.L. Dorroh appeared in 1932 and provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions such as the $A+ XB[X]$ and $A+ XB[[X]]$ constructions, the CPI-extensions of Boisen and Sheldon, the $D+M$ constructions and the Nagata's idealization.

math.AC↗