arXiv subjects

Jawad Abuhlail

Publications and source records attributed to Jawad Abuhlail.

At least 19 recordsLinked to original sources

arXiv2026

Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings

We study several separation axioms for $X$-top-lattices (i.e. a lattice $L$ for which a given subset $X\subseteq L\backslash \{1\}$ admits a \emph{% Zariski-like topology}). Such spaces are $T_{0}$ and usually far away from being $T_{2}.$ We provide sufficient/necessary conditions for an $X$-top lattice so that $X$ is $T_{2},$ \emph{regular} ($T_{3}$), \emph{completely regula}r ($T_{3\frac{1}{2}}$), \emph{normal}, \emph{completely normal} or \emph{perfectly normal} ($T_{6}$). We apply our results mainly to the spectrum of prime (resp. maximal, minimal) ideals of a commutative (semi)ring. We illustrate our results with several examples/counterexamples.

math.RA↗

arXiv2025

Flat Semimodules & von Neumann Regular Semirings

Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring.

math.RA↗

arXiv2019

PS-Hollow Representations of Modules over Commutative Rings

Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of \emph{pseudo strongly hollow submodules} (\emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with \emph{secondary representations}, we investigate modules which can be written as \emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of \emph{minimal} PS-hollow strongly representations of modules over Artinian rings.

math.AC↗

arXiv2019

Pushouts and e-Projective Semimodules

Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular the e-projective semimodules introduced by the first author using his new notion of exact sequences of semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive proof of their existence in a way that proved be very helpful.

math.RA↗

arXiv2019

On k-Noetherian and k-Artinian Semirings

We investigate left k-Noetherian and left k-Artinian semirings. We characterize such semirings using i-injective semimodules. We prove in particular, a partial version of the celebrated Bass-Papp Theorem for semiring. We illustrate our main results by examples and counter examples.

math.RA↗

arXiv2019

Injective Semimodules - Revisited

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreover, we provide partial results for the so called Embedding Problem (of semimodules in injective semimodules).

math.RA↗

arXiv2017

Second Representable Modules over Commutative Rings

Let $R$ be a commutative ring. We investigate $R$-modules which can be written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them \emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (\emph{main}) \emph{second attached prime ideals} related to a module with such a presentation.

math.AC↗

arXiv2017

Zariski-like Topologies for Lattices with Applications to Modules over Commutative Rings

We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so called classical Zariski topology $(X,τ^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be $\textit{spectral}$. We say that $\mathcal{L}$ is $X$\emph{-top} iff% \begin{equation*} τ:=\{X\backslash V(a)\mid a\in L\},\text{ where }V(a)=\{x\in L\mid a\leq x\} \end{equation*}% is a topology. We study the interplay between the \textit{algebraic properties} of an $X$-top complete lattice $\mathcal{L}$ and the $\textit{% topological properties}$ of $(X,τ^{cl})=(X,τ).$ Our results are applied to several spectra which are proper classes of $\mathcal{L}% :=LAT(_{R}M)$ where $M$ is a left module over an arbitrary associative ring $% R$ (e.g. the spectra of prime, coprime, fully prime submodules) of $M$ as well as to several spectra of the dual complete lattice $\mathcal{L}^{0}$ (e.g. the spectra of first, second and fully coprime submodules of $M$).

math.GN↗

arXiv2013

On Topological Lattices and an Application to First Submodules

We introduce the notion of a (strongly) topological lattice $\mathcal{L}=(L,\wedge ,\vee)$ with respect to a subset $X\subsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $\mathrm{Spec}^{\mathrm{f}}(M)$ of \textit{first submodules} of $M.$ We topologize $\mathrm{Spec}^{\mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the topological properties of the associated space.

math.RA↗

arXiv2013

Hopf Semialgebras

In this paper, we introduce and investigate \emph{bisemialgebras}and\emph{\ Hopf semialgebras} over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over rings including the main reconstruction theorems and the \emph{Fundamental Theorem of Hopf Algebras}. We also provide a notion of \emph{quantum monoids} as Hopf semialgebras which are neither commutative nor cocommutative; this extends the Hopf algebraic notion of a quantum group. The generalization to the semialgebraic context is neither trivial nor straightforward due to the non-additive nature of the base category of Abelian monoids which is also neither Puppe-exact nor homological and does not necessarily have enough injectives.

math.RA↗

arXiv2013

Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\boxtimes_{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\mathbb{S}%_{A},$ the base semialgebra $A$ has properties of a semiunit (in a sense which we clarify in this note). Motivated by this interesting example, we investigate semiunital semimonoidal categories $(\mathcal{V}%, \bullet, I)$ as a framework for studying notions like semimonoids (semicomonoids) as well as a notion of monads (comonads) which we call $\mathbb{J}$-monads ($\mathbb{J}$-% comonads) with respect to the endo-functor $\mathbb{J}:=\mathbf{I}\bullet -\simeq -\bullet \mathbf{I}:\mathcal{V}\longrightarrow \mathcal{V}.$ This motivated also introducing a more generalized notion of monads (comonads) in arbitrary categories with respect to arbitrary endo-functors. Applications to the semiunital semimonoidal variety $(_{A}\mathbb{S}%_{A},\boxtimes_{A},A)$ provide us with examples of semiunital $A$-semirings (semicounital $A$-semicorings) and semiunitary semimodules (semicounitary semicomodules) which extend the classical notions of unital rings (counital corings) and unitary modules (counitary comodules).

math.CT↗

arXiv2012

Exact Sequences of Semimodules over Semirings

In this paper, we introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties.

math.CT↗

arXiv2012

The dual notion of strong irreducibility

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

math.RA↗

arXiv2012

Uniformly Flat Semimodules

We revisit the notion of flatness for semimodules over semirings. In particular, we introduce and study a new notion of uniformly flat semimodules based on the exactness of the tensor functor. We also investigate the relations between this notion and other notions of flatness for semimodules in the literature.

math.RA↗

arXiv2011

Exact Sequences in Non-Exact Categories (An Application to Semimodules)

We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties to relative homological varieties.

math.CT↗

arXiv2011

Zariski Topologies for Coprime and Second Submodules

Let $M$ be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called \emph{second} and \emph{coprime} submodules of $M.$ Moreover, we topologize the spectrum $% \mathrm{Spec}^{\mathrm{s}}(M)$ of second submodules of $M$ and the spectrum $% \mathrm{Spec}^{\mathrm{c}}(M)$ of coprime submodules of $M,$ study several properties of these spaces and investigate their interplay with the algebraic properties of $M.$

math.RA↗

arXiv2010

A Zariski Topology for Modules

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration.

math.RA↗

arXiv2009

Tilting Modules over Almost Perfect Domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).

math.AC↗