Jawad Abuhlail

King Fahd University of Petroleum and Minerals | KFUPM · Department of Mathematics and Statistics

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called [Formula: see text]-injective semimodules introduced...

We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.

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...

We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative 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.

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...

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 aut...

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...

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 partic...

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective s...

Let R be a commutative ring. We investigate R-modules which can be written as finite sums of second R-submodules (we call them second representable). The class of second representable modules lies between the class of finitely generated semisimple modules and the class of representable modules; moreover, we give examples to show that these inclusio...

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 $(X,\tau ^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for i...

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-ze...

In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the b...

Using a restrictive notion of exactness and the natural tensor product, we generalize several results related to flat modules over rings to flat semimodules over semirings.

Basic homological lemmas well known for modules over rings and, more generally, in the context of abelian categories, have been extended to many other concrete and abstract-categorical contexts by various authors. We propose a new such extension, specifically for commutative monoids and semimodules; these two contexts are equivalent since the forge...

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 \...

In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is \emph{not} straightforward and even subtle at some places due to the nature of the base c...

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...

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 investig...

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 ex...

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.

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...

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some prelim-inaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prüfer conditions; namely, we prove that this class of rings stands...

We introduce a dual Zariski topology on the spectrum of fully coprime R-submodules of a given duo module M over an associative (not necessarily commutative) ring R. This topology is defined in a way dual to that of defining the Zariski topology on the prime spectrum of R. We investigate this topology and clarify the interplay between the properties...

Let M be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Specs(M) of second submodules of M and the spectrum Specc(M) of coprime submodules of M, study several properties of these spaces and investigate thei...

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...

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...

Given a coalgebra $C$ over a commutative ring $R,$ we show that $C$ can be considered as a (not necessarily counital) $C^{\ast op}$-coring. Moreover, we show that this coring has a left (right) counity if and only if $C$ is coseparable as an $R$-coalgebra.

An R-module M is a star-module if the functor Hom(M,-) induces an equivalence between the two categories Gen(M) and Cogen(M*_{S}) where S:=End(M) and M*:=\Hom_{R}(M,Q) for an injective cogenerator Q_{R}. This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is a star-module. We investigat...

This paper is an exposition of the so-called injective Morita contexts (in which the connecting bimodule morphisms are injective) and Morita $\alpha$contexts (in which the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-Huisgen). Motivated by situations in which only one trace ideal is in action, or the compatibility b...

In this paper we introduce and investigate top (bi)comodules} of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative rings. We restrict our attention in this...

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category o...

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case of locally projective corings as a composition of suitable ``Trace'' and ``Hom'' functors and show how to derive it from a m...

The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative gr...

In this note we study the weak topology on paired modules over a (not necessarily commutative) ground ring. Over QF rings we are able to recover most of the well known properties of this topology in the case of commutative base fields. The properties of the linear weak topology and the dense pairings are then used to characterize pairings satisfyin...

In this note we study dual coalgebras of algebras over arbitrary (noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called coreflexive coalgebras. Our approach in this note is an algebraically topological one.

In this note we introduce and investigate the concepts of dual entwining structures and dual entwined modules. This generalizes the concepts of dual Doi-Koppinen structures and dual Doi-Koppinen modules introduced (in the infinite case over rings) by the author is his dissertation.

We develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.

In this note we extend duality theorems for crossed products obtained by M. Koppinen and C. Chen from the case of a base field or a Dedekind domain to the case of an arbitrary noetherian commutative ground ring under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case...

In this note we study Morita contexts and Galois extensions for corings. For a coring $\QTR{cal}{C}$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\QTR{cal}{M}^{\QTR{cal}{C}}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called \QTR{em}{cleft}$C$\QTR{em}{-Galois exten...

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gomez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra...

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $...

C--comodules are isomorphic. This applies in particular for the canonical pairings (C, C # ) and (A # , A) derived from a coalgebra C and an algebra A, respectively. An attempt to develop systematically the theory of rational modules associated to a pairing (C, A), where C is a coalgebra and A is an algebra over an arbitrary commutative ring R, is...

In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (...

In dieser Arbeit übertragen wir Dualitätssätze für Hopf-Algebren vom Körper-Fall auf beliebige kommutative Grundringe unter schwachen und natürlichen Bedingungen. Insbesondere betrachten wir die kalssische Dualität zwischen den Gruppen und den kommutativen Hopf-Algebren und Dualitätssätze für die sogenannten verschränkten Produkte. Wir führen die K...

Düsseldorf, Universiẗat, Diss., 2001. Computerdatei im Fernzugriff.