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Let R be a commutative Noetherian ring, a proper ideal of R and N a nonzero finitely generated R -module with N ≠ N. Let d (respectively, c) be the smallest (respectively, greatest) non-negative integer i such that the local cohomology H i (N) is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules H d (N) , H c (N) and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch's conjecture. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the class of quasicomplemented distributive nearlattices. We investigate α-filters and α-ideals in quasicomplemented distributive nearlattices and some results on idealscongruence-kernels. Finally, we also study the notion of Stone distributive nearlattice and give a characterization by means σ-filters. [ABSTRACT FROM AUTHOR]

If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra $L_K(E)$ of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of $L_K(E)$ by ${\mathbb {Z}}$. Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital $L_K(E)$ having this property. [ABSTRACT FROM AUTHOR]

Let $\mathbb K$ be a field of characteristic zero, $A := \mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(\mathbb K)$ the Lie algebra of all $\mathbb K$-derivations on $A$. Every polynomial $f \in A$ defines a Jacobian derivation $D_f\in W_2(\mathbb K)$ by the rule $D_f(h)=\det J(f, h)$ for any $h\in A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(\mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(\mathbb K)$. It is proved that for a Jordan chain $T(f_1)=\lambda f_1+f_2$, ..., $T(f_{k-1})=\lambda f_{k-1}+f_k$, $T(f_k)=\lambda f_k$ for a derivation $T\in W_2(\mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$ in $W_2(\mathbb K)$. In case $A:=\mathbb K[x_1, \ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(\mathbb K)$ on the principal ideals $(f)$ is considered.

In the past few decades, researchers have found that studying modules using endomorphisms is a powerful and useful tool. This has led to valuable works in this field. Recently, the study of (dual) Rickart modules has become an important approach as they are deeply connected to endomorphisms. Building on this work, the authors introduce a new perspective on (dual) Rickart modules using isomorphism. We also define virtually (dual) Rickart modules. It is shown that rings with all modules virtually Rickart are semisimple rings. The paper includes various examples to illustrate the concepts presented.

In this paper, we investigate r-ideals in commutative ordered semigroups with zero and identity. Moreover, we study some properties of r-ideals. Furthermore, we give several characterizations of r-ideals. [ABSTRACT FROM AUTHOR]

We extend the concept of a G-Drazin inverse from the set M n of all n × n complex matrices to the set R D of all Drazin invertible elements in a ring R with identity. We also generalize a partial order induced by G-Drazin inverses from M n to the set of all regular elements in R D , study its properties, compare it to known partial orders, and generalize some known results. [ABSTRACT FROM AUTHOR]

We investigate general properties of multipliers and weak multipliers of algebras. We apply the results to determine the (weak) multipliers of associative algebras and zeropotent algebras of dimension 3 over an algebraically closed field. [ABSTRACT FROM AUTHOR]

The enhancement in the efficiency of triplet-triplet annihilation upconversion (TTA-UC) is mainly determined by the triplet energy transfer (TET) and triplet-triplet annihilation (TTA) between the sensitizers and annihilators. The TET process works efficiently by adjusting the concentration ratio of the sensitizers and annihilators. The efficiency of TTA is determined by the properties of the annihilator. Because TTA is a Dexter-type energy transfer and is affected by the diffusion rate, the energy levels of the excited states and the molecular size are both crucial in TTA. In this study, four isomerized dimers of 9,10-diphenlanthracene (DPA) and anthracene (An) were designed and prepared as annihilators for TTA-UC. The singlet and triplet energy levels could be adjusted by altering the connection position while maintaining the molecular weight and size. When PtOEP was used as the sensitizer, the maximum upconversion efficiency of 9-[4-(9-anthracenyl)phenyl]-10-phenylanthracene (9DPA-9An) was ∼11.18%. This is four times higher than that of 9,10-diphenyl-2,9 ' -bianthracene (2DPA-9An, 2.63%). The calculation of the energies of T1 and the higher triplet state (T3, because E (T2) is similar to the E (T1) of these dimers) for these dimers has provided insights into the underlying reasons. These indicated that the energy gap value of 2 × E (T1) - E (T3) is the determining factor for TTA efficiency. This work may provide a better understanding of the excited-state energy levels, which is crucial for designing novel annihilators to enhance the TTA-UC efficiency. [ABSTRACT FROM AUTHOR]

We give a complete combinatorial answer to Kostant's problem for simple highest weight modules indexed by fully commutative permutations. We also propose a reformulation of Kostant's problem in the context of fiab bicategories and classify annihilators of simple objects in the principal birepresentations of such bicategories generalizing the Barbasch-Vogan theorem for Lie algebras. [ABSTRACT FROM AUTHOR]

All automorphism of vector space V has the property that whenever V is embedded in a vector space W, can be extends to some automorphism of W. In a given category, characterizing the automorphisms having the extension property is a very difficult problem. Important results are published in the category of groups. Schupp (Proc Am Math Soc, 101:226-228, 1987) proved that the automorphisms satisfying the extension property in the category of groups, characterize the inner automorphisms. Then Pettet (Proc Am Math Soc 106:87-90) provided a simpler proof of Schupp's result. Later, Ben Yacoub (Portugaliae Mathematica 51:231-233, 1994) proved that Schupp's result (Schupp in Proc Am Math Soc, 101:226-228, 1987) cannot be extended to the category of algebras. In order to generalize the result of Schupp, Abdelalim and Essannouni (Portugaliae Mathematica, 59:325-334, 2002) characterized the automorphisms having the extension property, in the category of abelian groups. It was natural to study the extension property in module categories. In Abdelalim et al. (Rendiconti del Circolo Matematico di Palermo Series 2 1-6, 2022) and Abdelalim et al. (Ricerche di Matematica 1-15, 2023) characterized respectively, the automorphisms having the extension property, in the category of free modules over just an integral domain and a category of mixed modules over a bounded factorization domain. It is therefore logical to see what is happening in categories of torsion modules. Let M be a torsion direct finite sum of cyclic modules over a unique factorization domain A. Let α be an automorphism of the A-module M. This article aims to prove that α satisfies the extension property in the category of modules if and only if α is an homothety of invertible ratio in A. [ABSTRACT FROM AUTHOR]

Consider an assignment of bits to the vertices of a connected graph Γ (V , E) with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a storage code of length |V| on Γ . In this paper we construct an infinite family of linear storage codes on triangle-free graphs with rates arbitrarily close to one. [ABSTRACT FROM AUTHOR]

IntroductionVarious methods have been performed to control weeds in the world and the use of herbicides is one of them, but public concerns about human health have changed interest in alternative methods. Thermal methods based on flame-weeder, hot air, steam, and hot water have the potential to control weeds, but due to the high cost are not economical. Electromagnetic waves transfer energy into weeds and finally destroy them. The effect of radiation on plant mutation, high consumption of energy, and human health are problems for this approach. Unlike other methods, electrical energy is an ideal and non-chemical method for weeds. This method applies high voltage to weeds, their roots, and soil so that electric currents pass through them, and the vaporization of the liquid content of weeds kills the weeds. To increase the severity of damage to weeds, the development of a feedback mechanism is required. The ultrasonic sensor measuring physical parameters like plant height is a simple method. Some complex sensing systems include optical sensors such as infrared, and machine vision that require high-speed processors and expensive equipment. In this project, as a simple method, the monitoring of the electrical current passing through weeds was used for developing the feedback mechanism and increasing electric damage to weeds.Materials and MethodsIn this study, the system consisted of a high-voltage device that generated a 15 kV AC voltage to kill weeds, as well as a feedback mechanism that included a sensor to measure the electric current on the input of the weed killer and identify the presence of weeds and their annihilation. All parts were installed on a robotic platform, and an application on a laptop was connected to it via an access point for navigation and data reception. The system was tested in a greenhouse lab with various weeds. Initially, a test was performed to investigate the effect of high voltage on the weeds and establish relationships between the electric currents passing through weeds and their presence (before and after annihilation). During the test, the system was guided along a path and applied high voltage to kill the weeds. The feedback mechanism was then calibrated based on the extracted data on electric current relations. This allowed the system to detect weeds and their annihilation, enabling it to move to the next target once a weed had been eliminated. After calibration, a comparative test was conducted to evaluate the weed-killing efficiency of the two methods (with and without the feedback mechanism), and the results were analyzed using a t-test with p ≤ 0.01.Results and DiscussionThe observations indicated that the input electric current on the weed killer was dependent on the electric current passing through weeds. When the high-voltage electrode touched a weed, the electric current passed through it increased, and simultaneously, the high electrical energy destroyed the weed. After the removal of the weed, the electric current rapidly decreased. The average energy consumption per weed plant was estimated to be 250 joules, which can be compared with other methods. The final test comparing the use and non-use of the feedback mechanism revealed significant differences (P < 0.01) between the results obtained with and without the mechanism, demonstrating that the feedback mechanism increased the efficiency of weed annihilation. The sensing system used in the developed feedback mechanism is a simple method that is affected by the electrical resistivity of weeds. As such, it did not mistakenly detect other objects as weeds, unlike an ultrasonic mechanism. Based on these results, monitoring the electrical current passing through weeds proved to be a suitable method for developing a feedback mechanism for the weed killer to identify the presence of weeds and their annihilation.ConclusionThe use of high voltage as a non-chemical and alternative method for weed control has shown promising results. The study revealed that measuring the electric current applied to the weed killer was an effective and straightforward approach to developing a feedback mechanism. This mechanism aids in identifying the presence of weeds and ensuring their elimination by intensifying the damage inflicted on them through the application of high electrical energy. To further enhance the efficiency and speed of weed control, future research should consider integrating an automatic guidance mechanism with the weed killer.

This study aims to introduce the concept of α -ideal in bounded commutative residuated lattices and establish some related properties. In this paper, we show that the set of α -ideals of a bounded commutative residuated lattice is a Heyting algebra, and an algebraic lattice. Moreover, we state the prime α -ideal theorem, and describe relations between α -ideals and some types of ideals of a bounded commutative residuated lattice. Finally, we discuss correspondences between α -ideals and α -filters of a bounded commutative residuated lattice. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the annihilating-ideal graph of an MV-algebra (A, ⊕, ∗, 0). The algebraic structure of MV-algebras (especially Boolean algebras) are described by using the annihilating-ideal graph. The connections between the ideal theory of MV-algebras and graph theory are established, which promote the studying of the coloring of graphs. The annihilating-ideal graph AG(A) is a simple graph with the vertex set V (AG(A)) = {I ∈ I(A)\{h0i, A} | ∃J ∈ I ∗ (A) such that IJ = h0i} and the edge set E(AG(A)) = {I − J | IJ = h0i, where I, J ∈ V (AG(A)) and I 6= J}, where I(A) is the set of all ideals of A and I ∗ (A) = I(A)\{h0i}. We verify that AG(A) is connected with dmax(AG(A)) ≤ 3. And we characterize some MV-algebras with dmax(AG(A)) = 0 or 1, where dmax(AG(A)) is the diameter of AG(A). If | A |≤ 7, we show that AG(A) is either a null graph, or dmax(AG(A)) = 1. We restrict MV-algebras to Boolean algebras. The connections between AG(A) and Γ(A) are studied, where Γ(A) is the zero-divisor graph of A. We characterize the complete graph AG(A) and the star graph AG(A) by using ann(A\{1}) = {a ∈ A | a ⊙ b = 0 for all b ∈ A\{1}}, where ann(A\{1}) is the annihilator of A\{1}. Finally, we study the vertex coloring and girth of AG(A). We give two lower bounds and an upper bound for χ(AG(A)). [ABSTRACT FROM AUTHOR]

A d-dimensional configuration c : Z d ⟶ A is a coloring of the d-dimensional infinite grid by elements of a finite alphabet A ⊆ Z . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a (d - 1) -dimensional linear subspace S ⊆ R d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all (d - 1) -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of Z d by translations of a single tile. [ABSTRACT FROM AUTHOR]

Introduction: Various methods have been performed to control weeds in the world and the use of herbicides is one of them, but public concerns about human health have changed interest in alternative methods. Thermal methods based on flame-weeder, hot air, steam, and hot water have the potential to control weeds, but due to the high cost are not economical. Electromagnetic waves transfer energy into weeds and finally destroy them. The effect of radiation on plant mutation, high consumption of energy, and human health are problems for this approach. Unlike other methods, electrical energy is an ideal and non-chemical method for weeds. This method applies high voltage to weeds, their roots, and soil so that electric currents pass through them, and the vaporization of the liquid content of weeds kills the weeds. To increase the severity of damage to weeds, the development of a feedback mechanism is required. The ultrasonic sensor measuring physical parameters like plant height is a simple method. Some complex sensing systems include optical sensors such as infrared, and machine vision that require high-speed processors and expensive equipment. In this project, as a simple method, the monitoring of the electrical current passing through weeds was used for developing the feedback mechanism and increasing electric damage to weeds. Materials and Methods: In this study, the system consisted of a high-voltage device that generated a 15 kV AC voltage to kill weeds, as well as a feedback mechanism that included a sensor to measure the electric current on the input of the weed killer and identify the presence of weeds and their annihilation. All parts were installed on a robotic platform, and an application on a laptop was connected to it via an access point for navigation and data reception. The system was tested in a greenhouse lab with various weeds. Initially, a test was performed to investigate the effect of high voltage on the weeds and establish relationships between the electric currents passing through weeds and their presence (before and after annihilation). During the test, the system was guided along a path and applied high voltage to kill the weeds. The feedback mechanism was then calibrated based on the extracted data on electric current relations. This allowed the system to detect weeds and their annihilation, enabling it to move to the next target once a weed had been eliminated. After calibration, a comparative test was conducted to evaluate the weed-killing efficiency of the two methods (with and without the feedback mechanism), and the results were analyzed using a t-test with p ≤ 0.01. Results and Discussion: The observations indicated that the input electric current on the weed killer was dependent on the electric current passing through weeds. When the high-voltage electrode touched a weed, the electric current passed through it increased, and simultaneously, the high electrical energy destroyed the weed. After the removal of the weed, the electric current rapidly decreased. The average energy consumption per weed plant was estimated to be 250 joules, which can be compared with other methods. The final test comparing the use and non-use of the feedback mechanism revealed significant differences (P < 0.01) between the results obtained with and without the mechanism, demonstrating that the feedback mechanism increased the efficiency of weed annihilation. The sensing system used in the developed feedback mechanism is a simple method that is affected by the electrical resistivity of weeds. As such, it did not mistakenly detect other objects as weeds, unlike an ultrasonic mechanism. Based on these results, monitoring the electrical current passing through weeds proved to be a suitable method for developing a feedback mechanism for the weed killer to identify the presence of weeds and their annihilation. Conclusion: The use of high voltage as a non-chemical and alternative method for weed control has shown promising results. The study revealed that measuring the electric current applied to the weed killer was an effective and straightforward approach to developing a feedback mechanism. This mechanism aids in identifying the presence of weeds and ensuring their elimination by intensifying the damage inflicted on them through the application of high electrical energy. To further enhance the efficiency and speed of weed control, future research should consider integrating an automatic guidance mechanism with the weed killer. [ABSTRACT FROM AUTHOR]

As is known, for isolated hypersurface singularities and complete intersections of positive dimension, the Milnor number is the least upper bound for the Tjurina number, i.e., . In this paper we show that, for zero-dimensional complete intersections, the reverse inequality holds. The proof is based on properties of faithful modules over an Artinian local ring. We also exploit simple properties of the annihilator and the socle of the modules of Kähler differentials and derivations and the theory of duality in the cotangent complex of zero-dimensional singularities. [ABSTRACT FROM AUTHOR]

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$. As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs ['A characterisation result for matrix rings', Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R , we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so. [ABSTRACT FROM AUTHOR]

The annihilating-ideal graph of a commutative ring R with unity is deffned as the graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect. [ABSTRACT FROM AUTHOR]

The annihilating-ideal graph of a commutative ring R with unity is deffned as the graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect. [ABSTRACT FROM AUTHOR]

A ring R is called a left P S -ring if its left socle, Soc (R R) , is projective. Equivalently, R is left P S -ring if the right annihilator of every maximal left ideal is of the form e R , where e is an idempotent in R. In this paper, we characterize when a (trivial) Morita context or a generalized triangular matrix ring is a left P S -ring. Examples to illustrate and delimit the theory are provided. [ABSTRACT FROM AUTHOR]

In this paper we describe commutative monoids S containing a zero element in which every ideal is the annihilator of an element of S. [ABSTRACT FROM AUTHOR]

This paper investigates uninorms that are neither conjunctive nor disjunctive on bounded lattices. New methods are introduced for construction of such uninorms, where some restrictions on the identity and the annihilator are considered. In particular, new types of idempotent uninorms on bounded lattices are obtained. Furthermore, some specific examples are provided to illustrate that these constructions differ from the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we transfer Davey‘s characterization for κ -Stone bounded distributive lattices to lattices with certain kinds of quotients, in particular to commutator lattices with certain properties, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of these lattices. We then apply these results to certain congruence lattices, in particular to those of semiprime members of semi-degenerate congruence-modular varieties, and use this particular case to transfer Davey‘s Theorem to commutative unitary rings.

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph Γ′I(R) and prove that Γ′I(R) is connected with diameter at most two and if Γ′I(R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph Γ′I(R) and the ideal-based zero-divisor graph ΓI (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph Γ′I(R) is identical to the ideal-based zero-divisor graph ΓI (R) if and only if R has exactly two minimal prime-ideals which contain I. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic p and let Ω n (F) be the F -vector space of n -differential forms. In this work, we will study the annihilator of differential forms, give specific descriptions for special cases and show a connection between these annihilators and the kernels of the restriction map Ω n (F) → Ω n (E) for purely inseparable field extensions E / F. [ABSTRACT FROM AUTHOR]

We define the notions of annihilators in hoops and study some related attributes of them, and we show that annihilators are ideals of hoop. In addition, we show that all ideals of hoop are a bounded distributive pseudo-complement lattice, and we use this result and demonstrate that the collection of all annihilators of hoop is a Boolean algebra. Also, we use the notion of annihilator and introduce the special kind of ideal of hoop as ς -ideal and display that all ς -ideals of hoop are a complete distributive lattice and we consequence that under what condition it is a Boolean algebra. [ABSTRACT FROM AUTHOR]

We study the following two conditions in rings: (i) the right annihilator of some power of any element is an ideal, and (ii) the right annihilator of any nonzero element a contains an ideal generated by some power of any right zero-divisor of the element a. We investigate the structure of rings in relation to these conditions; especially, a ring with the condition (ii) is called right APIP. These conditions are shown to be not right-left symmetric. For a prime two-sided APIP ring R we prove that every element of R is either nilpotent or regular, and that if R is of bounded index of nilpotency then R is a domain. We also provide several interesting examples which delimit the classes of rings related to these properties. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to characterize two wide classes of nullnorms on bounded lattices. First, we construct two methods of nullnorms on bounded lattices and find sufficient and necessary conditions for nullnorms that possess an annihilator. Moreover, we prove that the conditions are equivalent and illustrate them with examples. Finally, we present another two nullnorms on bounded lattices and illustrate that they differ from nullnorms constructed by Çaylı in 2020. [ABSTRACT FROM AUTHOR]

The notion of divisibility is introduced in a distributive lattice with respect to a filter and it is proved that the set of all multipliers of an element is a filter. A congruence is defined on a distributive lattice with respect to these multiplier filters and established a set of equivalent conditions for the corresponding quotient lattice to become a Boolean algebra. The concepts of D-prime elements and D-irreducible elements are introduced and characterized in terms of corresponding multiplier filters. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, M be a unitary R -module and Γ be a simple connected graph. We examine different equivalence relations on subsets A f (M) \ { 0 } , A s (M) \ { 0 } and A t (M) \ { 0 } of M , where A f (M) is the set of full-annihilators, A s (M) is the set of semi-annihilators and A t (M) is the set of star-annihilators in M. We prove that elements x , y ∈ M are neighborhood similar in the annihilating graph ann f (Γ (M)) if and only if the submodules ann (x) M and ann (y) M of M are equal. We study the isomorphism of annihilating graphs arising from M and the tensor product M ⊗ R T − 1 R , where T = R \ C (M) , C (M) = { r ∈ R   ∣   r m = 0 for some 0 ≠ m ∈ M }. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a Z -module). The main objective of this paper is to study properties of mod-annihilators of M. For x ∈ M , we study the ideals [ x : M ] = { r ∈ R | r M ⊆ R x } of R corresponding to mod-annihilator of M. We investigate as when [x : M] is an essential ideal of R. We prove that the arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that [x : M] is injective if and only if R is non-singular and the radical of R/[x : M] is zero. Moreover, if essential socle of M is non-zero, then we show that [x : M] is the intersection of maximal ideals and [ x : M ] 2 = [ x : M ] . Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R. [ABSTRACT FROM AUTHOR]

So far, there are no efficient algorithms to solve a problem of finding the low degree annihilators for arbitrary Boolean function. In the paper we present a new algorithm to find affine annihilators for an arbitrary Boolean function. We start with considering the identity fg ≡ 0 or the arbitrary Boolean function f and its possible affine annihilator g. We use a special representation of the Boolean function in sum of its sub-functions to reduce degrees of considering functions in previous identity. As a result, we establish equivalence between the identity fg ≡ 0 for Boolean functions of n variables and the system of Boolean equations of n-1 variable.An algorithm for finding the affine annihilators for the arbitrary Boolean function f must find all the affine functions g so that fg ≡ 0. Our algorithm is based on reducing the problem of finding the affine annihilators for the Boolean function f of n to the similar problem for its sub-functions of n-1 variable. The presented algorithm has the following advantages:An input function can be presented in different ways;Output can be also presented in different ways;The algorithm can be effectively parallelized.It should be noted that the result we have obtained is not final and highlights some development directions: first, to study the impact of its input and output on the efficiency of the algorithm of various representations and, second, to use our idea of constructing the algorithm for development of algorithms, which allow finding the 2nd, 3rd, etc. degree annihilators for a specified Boolean function.

In the article we study boolean functions with affine annihilators. We have obtained results in both, estimating the number of functions under study and defining the relationship between Walsh-Hadamard coefficients of an arbitrary boolean function and its affine annihilator available. The second section of this article focuses on estimating the number of boolean functions with affine annihilators. The value has top and bottom bound. Besides, we have obtained the asymptotic estimate of the number of boolean functions with affine annihilators. The third section studies the Walsh-Hadamard coefficients of boolean functions with affine annihilators. First, we have derived the dependence of the Walsh-Hadamard coefficient on the distance between an arbitrary boolean function and a vector space of the affine function’s annihilators. Based on this result, we have obtained the dependence of distance between an arbitrary boolean function and a set of functions with affine annihilators on the spectrum of given function. Also we have defined the necessary and sufficient condition for the arbitrary boolean function to be with an affine annihilator available. Using the results obtained we bounded an absolute value of Walsh-Hadamard coefficients.Also we suggested a method for boolean equations analysis, which is based on two known methods. Namely, we used an analysis using annihilators and an analysis using linear analogs. We have obtained an estimate of the success probability of the suggested method for an arbitrary boolean function. Also we proved that bent functions are the most resistant to this analysis.The results obtained can be used in analysis of boolean equations. Also obtained dependences can be used, for instance, to study bent functions and algebraic immunity of boolean functions.

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated. [ABSTRACT FROM AUTHOR]

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C (X) to characterize spaces X when every irreducible ideal of C (X) is pseudoprime. We also characterize the rings of continuous functions C (X) in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an Archimedean reduced ring and satisfy ACC on annihilators, then R [ [ x ] ] is also an Archimedean reduced ring. More generally, we prove that if R is a right Archimedean ring satisfying the ACC on annihilators and α is a rigid automorphism of R , then the skew power series ring R [ [ x ; α ] ] is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a commutative Noetherian ring R and let M be a finitely generated R-module. The purpose of this paper is to establish some new results on Lynch's conjecture and the set of the attached primes of the top local cohomology module H a cd (a , M) (M) for the case that R has prime characteristic and cd (a , M) = ara M (a) . To prove our results, we establish and use a new relation between the set Att R (H a cd (a , M) (M)) and the M-height of the annihilator of the top local cohomology module H a cd (a , M) (M) . Several corollaries of this result are proved. Among other things, we will provide a complete characterization of the attached primes of the top local cohomology module H a cd (a , M) (M) and Rad (Ann R (H a cd (a , M) (M))) . Using this, we show that the ideal Ann R (H a cd (a , M) (M)) has M-height zero. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a commutative Noetherian ring R and t be a nonnegative integer. Let M and N be two finitely generated R-modules. In certain cases, we give some bounds under inclusion for the annihilators of ExtRt(M, N) and H a t (M) in terms of minimal primary decomposition of the zero submodule of M, which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases. [ABSTRACT FROM AUTHOR]

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Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in R[[X]], then it will holds in R.

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first point out some mistakes in [12]. Especially the Theorem 3.9 [12] showed that: Let A be residuated lattice and ∅ ≠ X ⊆ A, then the least ideal containing X can be expressed as: 〈X〉 = {a ∈ A|a ≤ (·· · ((x1 ⊕ x2) ⊕ x3) ⊕ ·· ·) ⊕ xn, xi ∈ X, i = 1, 2 ·· · , n}. But we present an example to illustrate the ideal generation formula may not hold on residuated lattices. Further we give the correct ideal generation formula on residuated lattices. Moreover, we extend the concepts of annihilators and α-ideals to MTL-algebras and focus on studying the relations between them. Furthermore, we show that the set Iα (M) of all α-ideals on a linear MTL-algebra M only contains two trivial α-ideals {0} and M. However, the authors [24] studied the structure of Iα (M) in a linear BL-algebra M, which means some results with respect to Iα (M) given in [24] are trivial. Unlike that, we investigate the lattice structure of Iα (M) on general MTL-algebras. [ABSTRACT FROM AUTHOR]

In this short paper we introduce the notion of $\alpha$-filter in the class of distributive nearlattices and we prove that the $\alpha$-filters of a normal distributive nearlattice are strongly connected with the filters of the distributive nearlattice of the annihilators.

In this note we give some new characterizations of distributivity of a nearlattice and we study annihilator-preserving congruence relations.

In this paper, the annihilator of an arbitrary subset of a pseudo BCI-algebra is defined and some related properties are studied. Applying the annihilator concept, some necessary and sufficient conditions for a pseudo BCI-algebra to be p-semisimple are established. Also, the normal pseudo BCI-ideal is defined and studied. The relation between normal pseudo BCI-ideals and pseudo BCK-algebras is discussed. Finally, the involutory pseudo BCI-ideal is defined and it is shown that if {0} is a normal pseudo BCI-ideal then the set of all involutory pseudo BCI-ideals forms a complete and complemented lattice. [ABSTRACT FROM AUTHOR]

Given elements a, b, c of any associative ring R with 1, then a is called right hybrid (b, c)-invertible if there exists y ∈ R such that yay = y , y R = b R and rann (y) = rann (c). It is shown that such y exists if and only if c ∈ cabR and rann (cab) ⊆ rann (b) , in which case y is unique. With an appropriate generalization of the right annhilator rann (.), this result is extended to all semigroups with 1 and to all categories. The semigroup version of right (and left) annihilator ideals is also used to define five finiteness properties for semigroups, and to establish the implications between them. Corresponding results for categories are also obtained. [ABSTRACT FROM AUTHOR]

The study of algebraic systems using graphs gives many interesting results. The ternary algebraic structures can be dealt with 3-uniform hypergraphs in which hyperedges are of size three. Right ternary near-ring, a generalization of near-ring in ternary context, was introduced by Daddi and Pawar in 2011. In this paper, annihilator 3-uniform hypergraph associated with the right ternary near-ring N denoted by AH3(N) is introduced. AH3(N) is seen to be empty when Nis a constant RTNR and it is complete when N is a zero RTNR. If N is integral, then the nature of AH3(N) is studied. A necessary condition for AH3(N) to be complete is derived. Hypergraph invariants of AH3(Zn) are obtained. For certain RTNR, the existence of BIBD is verified. [ABSTRACT FROM AUTHOR]