Jawad Abuhlail

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Publications (8)0.48 Total impact

  • Article: On the notion of strong irreducibility and its dual
    Jawad Abuhlail, Christian Lomp
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    ABSTRACT: 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.
    Journal of Algebra and Its Applications 11/2012; · 0.48 Impact Factor
  • Article: Exact Sequences of Semimodules over Semirings
    Jawad Abuhlail
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    ABSTRACT: 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.
    10/2012;
  • Article: Semiunital Semimonoidal Categories (Applications to Semirings and Semicorings)
    Jawad Abuhlail
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    ABSTRACT: 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).
    09/2012;
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    Article: Uniformly Flat Semimodules
    Jawad Abuhlail
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    ABSTRACT: 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.
    01/2012;
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    Article: Exact Sequences in Non-Exact Categories (An Application to Semimodules)
    Jawad Abuhlail
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    ABSTRACT: 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.
    11/2011;
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    Article: Zariski Topologies for Coprime and Second Submodules
    Jawad Abuhlail
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    ABSTRACT: 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 $M.$
    08/2010;
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    Article: A Zariski Topology for Modules
    Jawad Abuhlail
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    ABSTRACT: 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. Comment: 22 pages; submitted
    07/2010;
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    Article: Tilting Modules over Almost Perfect Domains
    Jawad Abuhlail, Mohammad Jarrar
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    ABSTRACT: 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).
    04/2009;
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