In this paper we investigate in details derivations on trivial extension algebras. We obtain gene... more In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.
In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan deriv... more In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in [S. Ali and A. Fošner, On Generalized (m, n)-Derivations and Generalized (m, n)-Jordan Derivations in Rings, Algebra Colloq. 21 (2014), 411-420] and [A. Fošner, A note on generalized (m, n)-Jordan cen- tralizers, Demonstratio Math. 46 (2013), 254-262]. Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer).
It is known that every nonzero Jordan ideal of 2-torsion free semiprime rings contains a nonzero ... more It is known that every nonzero Jordan ideal of 2-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a 2-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, 16 (2015), 769 -779] to the context of nonzero left ideals.
The notion of trivial extension of a ring by a module has been extensively studied and used in ri... more The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring R and a family M = (M i ) n i=1 of n R-modules for a given integer n ≥ 1. We call this new ring construction an n-trivial extension of R by M . In particular, the classical trivial extension will be just the 1-trivial extension. Thus we generalize several known results on the classical trivial extension to the setting of n-trivial extensions and we give some new ones. Various ring-theoretic constructions and properties of n-trivial extensions are studied and a detailed investigation of the graded aspect of n-trivial extensions is also given. We end the paper with an investigation of various divisibily properties of n-trivial extensions. In this context several open questions arise.
In this paper, we investigate Jordan ?-derivations and Lie ?-derivations on path algebras. This w... more In this paper, we investigate Jordan ?-derivations and Lie ?-derivations on path algebras. This work is motivated by the one of Benkovic done on triangular algebras and the study of Jordan derivations and Lie derivations on path algebras done by Li and Wei. Namely, main results state that every Jordan ?-derivation is a ?-derivation and every Lie ?-derivation is of a standard form on a path algebra when the associated quiver is acyclic and finite.
In this paper, we investigate the problem of describing the form of Lie generalized derivations o... more In this paper, we investigate the problem of describing the form of Lie generalized derivations on trivial extension algebras. We show, under some conditions, that every Lie generalized derivation on a trivial extension algebra is an sum of a generalized derivation and a center valued map which vanishes on all commutators. As an application we characterize Lie generalized derivation on a triangular algebra.
Let R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weak... more Let R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever {a,b,c\in R} and {0\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I o...
Fluoride ions are necessary and beneficial for human beings, however high levels of fluoride in g... more Fluoride ions are necessary and beneficial for human beings, however high levels of fluoride in groundwater can be toxic and harmful to human health and water resources, causing serious problems on teeth and bones. This situation is widespread in many parts of the world, including Morocco. According to World Health Organization (WHO), the maximum allowable concentration of fluoride in drinking water is 1.5 mg/L. The objective of this work is to understand the mechanism of transfer of fluoride ions in nanofiltration (NF) membranes. Two methodologies have been used, one mathematical, Nernst-Planck equation combined with film theory (NP-FT), and the other statistical artificial neural network (ANN). This study has been realized with experimental data obtained from three NF membranes (TR60, NF270 and NF90) that have been used for the removal of fluoride from NaF-doped groundwater with different initial concentrations of 5, 10, 15 and 20 mg/L. In a first part, the NP-FT model that is proposed as a contribution to the modeling of the concentration polarization phenomenon and to study the influence of the initial concentration (C i) of fluoride on the reflection coefficient (σ), the permeability of the solute (P s) and the thickness of the boundary layer (δ) has been made. In a second part, the two models ANN and NP-FT are used to predict the fluoride rejection as a function of the permeate flux. In addition, a comparison is made in terms of coefficient of determination R 2 between the two models. The results obtained showed that the influence of concentration bias is more important for NF270 than for TR60 and NF90, and high reflection coefficients, almost equal to unity, were obtained for NF90, indicating that convective transport is almost completely hindered. Similarly, reflection coefficients of about 0.97 and 0.89 were found for TR60 and NF270, respectively, suggesting that both mechanisms are present, but that diffusion is more pronounced for NF270 than for TR60. Thus, the thickness of the boundary layer and permeability change inversely to one another. It is about 10-5 m for NF270 and 10-6 m for TR60. On the other hand, the value obtained for the NF90 membrane is 10-14 m which is very low. The obtained solute permeability follows the order: R60 > NF270 > NF90. Although the experimental data agree well with the data predicted by ANN and NP-FT, which indicates that both models are appropriate for the prediction and satisfactory R 2. However, the ANN model reveals a slight superiority over the NP-FT model.
In this paper we investigate in details derivations on trivial extension algebras. We obtain gene... more In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.
The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear ma... more The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f-biderivations as a natural "biderivation" counterpart of the notion of "f-derivations". We first show, under some conditions, that any f-derivation is a Jordan biderivation. Then, we turn to study f-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence we show that every Jordan biderivation on a triangular algebra is a biderivation.
In this paper, we investigate Lie generalized derivations on bound quiver algebras associated wit... more In this paper, we investigate Lie generalized derivations on bound quiver algebras associated with a finite acyclic quiver. The first main result shows that every bound quiver algebra associated with a finite acyclic quiver has the properness Lie generalized derivation property. The second main result investigates the uniqueness property. Namely, among other things, we show that a bound quiver algebra associated with a finite acyclic quiver E has the uniqueness Lie generalized derivation property if and only if the quiver E does not contain any isolated vertex.
In this paper we prove, under certain condition, when R is a semiprime ring with suitable charact... more In this paper we prove, under certain condition, when R is a semiprime ring with suitable characteristic restrictions, that every nonzero (m,n)-Jordan triple centralizer (resp., (m,n)-Jordan triple derivation) is a two-sided centralizer (resp., a derivation which maps R into Z(R)). This give partial affirmative answers to two conjectures of Vukman.
In this paper, we introduce the notions of rough n-absorbing (n-absorbing primary) ideals and rou... more In this paper, we introduce the notions of rough n-absorbing (n-absorbing primary) ideals and rough fuzzy n-absorbing (n-absorbing primary) ideals in a ring, and give some properties of such ideals. Also, we discuss the relations between the upper and lower rough n-absorbing (n-absorbing primary) ideals and the upper and lower approximations of their homomorphism images.
In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan deri... more In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fo\v{s}ner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp., a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided centralizer).
In this paper we investigate in details derivations on trivial extension algebras. We obtain gene... more In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.
In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan deriv... more In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in [S. Ali and A. Fošner, On Generalized (m, n)-Derivations and Generalized (m, n)-Jordan Derivations in Rings, Algebra Colloq. 21 (2014), 411-420] and [A. Fošner, A note on generalized (m, n)-Jordan cen- tralizers, Demonstratio Math. 46 (2013), 254-262]. Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer).
It is known that every nonzero Jordan ideal of 2-torsion free semiprime rings contains a nonzero ... more It is known that every nonzero Jordan ideal of 2-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a 2-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, 16 (2015), 769 -779] to the context of nonzero left ideals.
The notion of trivial extension of a ring by a module has been extensively studied and used in ri... more The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring R and a family M = (M i ) n i=1 of n R-modules for a given integer n ≥ 1. We call this new ring construction an n-trivial extension of R by M . In particular, the classical trivial extension will be just the 1-trivial extension. Thus we generalize several known results on the classical trivial extension to the setting of n-trivial extensions and we give some new ones. Various ring-theoretic constructions and properties of n-trivial extensions are studied and a detailed investigation of the graded aspect of n-trivial extensions is also given. We end the paper with an investigation of various divisibily properties of n-trivial extensions. In this context several open questions arise.
In this paper, we investigate Jordan ?-derivations and Lie ?-derivations on path algebras. This w... more In this paper, we investigate Jordan ?-derivations and Lie ?-derivations on path algebras. This work is motivated by the one of Benkovic done on triangular algebras and the study of Jordan derivations and Lie derivations on path algebras done by Li and Wei. Namely, main results state that every Jordan ?-derivation is a ?-derivation and every Lie ?-derivation is of a standard form on a path algebra when the associated quiver is acyclic and finite.
In this paper, we investigate the problem of describing the form of Lie generalized derivations o... more In this paper, we investigate the problem of describing the form of Lie generalized derivations on trivial extension algebras. We show, under some conditions, that every Lie generalized derivation on a trivial extension algebra is an sum of a generalized derivation and a center valued map which vanishes on all commutators. As an application we characterize Lie generalized derivation on a triangular algebra.
Let R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weak... more Let R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever {a,b,c\in R} and {0\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I o...
Fluoride ions are necessary and beneficial for human beings, however high levels of fluoride in g... more Fluoride ions are necessary and beneficial for human beings, however high levels of fluoride in groundwater can be toxic and harmful to human health and water resources, causing serious problems on teeth and bones. This situation is widespread in many parts of the world, including Morocco. According to World Health Organization (WHO), the maximum allowable concentration of fluoride in drinking water is 1.5 mg/L. The objective of this work is to understand the mechanism of transfer of fluoride ions in nanofiltration (NF) membranes. Two methodologies have been used, one mathematical, Nernst-Planck equation combined with film theory (NP-FT), and the other statistical artificial neural network (ANN). This study has been realized with experimental data obtained from three NF membranes (TR60, NF270 and NF90) that have been used for the removal of fluoride from NaF-doped groundwater with different initial concentrations of 5, 10, 15 and 20 mg/L. In a first part, the NP-FT model that is proposed as a contribution to the modeling of the concentration polarization phenomenon and to study the influence of the initial concentration (C i) of fluoride on the reflection coefficient (σ), the permeability of the solute (P s) and the thickness of the boundary layer (δ) has been made. In a second part, the two models ANN and NP-FT are used to predict the fluoride rejection as a function of the permeate flux. In addition, a comparison is made in terms of coefficient of determination R 2 between the two models. The results obtained showed that the influence of concentration bias is more important for NF270 than for TR60 and NF90, and high reflection coefficients, almost equal to unity, were obtained for NF90, indicating that convective transport is almost completely hindered. Similarly, reflection coefficients of about 0.97 and 0.89 were found for TR60 and NF270, respectively, suggesting that both mechanisms are present, but that diffusion is more pronounced for NF270 than for TR60. Thus, the thickness of the boundary layer and permeability change inversely to one another. It is about 10-5 m for NF270 and 10-6 m for TR60. On the other hand, the value obtained for the NF90 membrane is 10-14 m which is very low. The obtained solute permeability follows the order: R60 > NF270 > NF90. Although the experimental data agree well with the data predicted by ANN and NP-FT, which indicates that both models are appropriate for the prediction and satisfactory R 2. However, the ANN model reveals a slight superiority over the NP-FT model.
In this paper we investigate in details derivations on trivial extension algebras. We obtain gene... more In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.
The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear ma... more The notion of f-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f-biderivations as a natural "biderivation" counterpart of the notion of "f-derivations". We first show, under some conditions, that any f-derivation is a Jordan biderivation. Then, we turn to study f-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence we show that every Jordan biderivation on a triangular algebra is a biderivation.
In this paper, we investigate Lie generalized derivations on bound quiver algebras associated wit... more In this paper, we investigate Lie generalized derivations on bound quiver algebras associated with a finite acyclic quiver. The first main result shows that every bound quiver algebra associated with a finite acyclic quiver has the properness Lie generalized derivation property. The second main result investigates the uniqueness property. Namely, among other things, we show that a bound quiver algebra associated with a finite acyclic quiver E has the uniqueness Lie generalized derivation property if and only if the quiver E does not contain any isolated vertex.
In this paper we prove, under certain condition, when R is a semiprime ring with suitable charact... more In this paper we prove, under certain condition, when R is a semiprime ring with suitable characteristic restrictions, that every nonzero (m,n)-Jordan triple centralizer (resp., (m,n)-Jordan triple derivation) is a two-sided centralizer (resp., a derivation which maps R into Z(R)). This give partial affirmative answers to two conjectures of Vukman.
In this paper, we introduce the notions of rough n-absorbing (n-absorbing primary) ideals and rou... more In this paper, we introduce the notions of rough n-absorbing (n-absorbing primary) ideals and rough fuzzy n-absorbing (n-absorbing primary) ideals in a ring, and give some properties of such ideals. Also, we discuss the relations between the upper and lower rough n-absorbing (n-absorbing primary) ideals and the upper and lower approximations of their homomorphism images.
In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan deri... more In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fo\v{s}ner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivation (resp., a generalized $(m,n)$-Jordan centralizer) is a derivation (resp., a two-sided centralizer).
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