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Let $\mathcal{L}$ be a bounded lattice. The essential element graph of $\mathcal{L}$ is a simple undirected graph $\varepsilon_{\mathcal{L}}$ such that the elements $x,y$ of $\mathcal{L}$ form an edge in $\varepsilon_{\mathcal{L}}$, whenever $x \vee y $ is an essential element of $\mathcal{L}$. In this paper, we study properties of the essential elements of lattices and essential element graphs. We study the lattices whose zero-divisor graphs and incomparability graphs are isomorphic to its essential element graphs. Moreover, the line essential element graphs are investigated.

The authors are supported by the Junta de Andalucía through projects FQM-336 and UMA18-FEDERJA-119 and by the Spanish Ministerio de Ciencia e Innovación through project PID2019-104236GB-I00, all of them with FEDER funds. It is known that the ideals of a Leavitt path algebra LK(E) generated by Pl(E), by Pc(E), or by Pec(E) are invariant under isomorphism. Though the ideal generated by Pb∞(E) is not invariant we find its "natural" replacement (which is indeed invariant): the one generated by the vertices of Pb∞p (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the DCC topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if H is a hereditary saturated subset of vertices providing an invariant ideal, its exterior ext(H) in the DCC topology of E0 generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance Pl, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain Soc(1)( ) ⊆ Soc(2)( ) ⊆ · · · , all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{Ann}} (R) $, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if$ I \cap \mathrm{Ann} _{R} (J) \neq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I) \neq \lbrace 0\rbrace $.In this paper, all rings with planar annihilator ideal graphs are classified.Furthermore, we show that all annihilator ideal graphs are perfect. Among other results, it is proved that if $\Gamma _{\mathrm{Ann}} (R) $ is a tree, then $\Gamma _{\mathrm{Ann}} (R) $ is star.

In [Linear Algebra Appl. 436 (2012), 1909–1923], the author introduced, for any semigroup S and any a,b,c∈S, the (b, c)-invertibility of a as meaning the existence of some (b, c)-inverse y∈S such t...

Let G be a finite group and k a finite field whose characteristic divides the order of G . Recall that an associated prime of H ⁎ ( G , k ) is a prime ideal of H ⁎ ( G , k ) which is the annihilator of some non-zero element of H ⁎ ( G , k ) , and that depth H ⁎ ( G , k ) refers to the maximal length of a regular sequence in H ⁎ ( G , k ) . J.F. Carlson conjectured that depth H ⁎ ( G , k ) = min ⁡ { dim ⁡ p } where p ranges over the associated primes of H ⁎ ( G , k ) . We show that for any finitely generated connected k -algebra A ( A n = 0 for n 0 , A 0 = k ) and for any finitely generated A -module M , the statement depth M = min ⁡ { dim ⁡ p | p an associated prime of M } is equivalent to several other statements concerning the module M , in particular that there exists a non-zero A -homomorphism M ∨ → H m d M where M ∨ is the k -dual of M and d is the depth of M . Then a spectral sequence due to Greenlees shows this last condition is true when the edge homomorphism in this spectral sequence is non-trivial. This is trivially true if the spectral sequence collapses, which happens when dim ⁡ H ⁎ ( G , k ) = depth H ⁎ ( G , k ) + 1 .

For a connected graph G(V, E) a set of vertices $$S\subseteq V(G)$$ resolves the graph G, and S is a resolving set of G, if every vertex is uniquely determined by its vector of distances to the vertices of S. A resolving set S of minimum cardinality is a metric basis for G, and the number of elements in the resolving set of minimum cardinality is the metric dimension of G. Let R be a commutative ring with non-zero identity. The annihilator graph of R, denoted by AG(R), is the (undirected) graph whose vertex set is the set of all non-zero zero-divisors of R and two distinct vertices x and y are adjacent if and only if $$ann_R(xy)\ne ann_R(x)\cup ann_R(y)$$ . In this paper, the metric dimension of the complement of AG(R) is studied and some metric dimension formulae for this graph are given.

In this paper, we introduce the concept of residual quotient of intuitionistic fuzzy subsets of ring and module and then define the notion of residual quotient intuitionistic fuzzy submodules , residual quotient intuitionistic fuzzy ideals. We study many properties of residual quotient relating to union, intersection,sum of intuitionistic fuzzy submodules (ideals). Using the concept of residual quotient, we investigate some important characterization of intuitionistic fuzzy annihilator of subsets of ring and module. We also study intuitionistic fuzzy prime submodules with the help of intuitionistic fuzzy annihilators. Many related properties are defined and discussed

In the strive to develop triplet-triplet annihilation photon upconversion (TTA-UC) to become applicable in a viable technology, there is a need to develop upconversion systems that can function well in solid states. One method to achieve efficient solid-state TTA-UC systems is to replace the intermolecular energy-transfer steps with the corresponding intramolecular transfers, thereby minimizing loss channels involved in chromophore diffusion. Herein, we present a study of photon upconversion by TTA internally within a polymeric annihilator network (iTTA). By the design of the annihilator polymer and the choice of experiment conditions, we isolate upconversion emission governed by iTTA within the annihilator particles and eliminate possible external TTA between separate annihilator particles (xTTA). This approach leads to mechanistic insights into the process of iTTA and makes it possible to explore the upconversion kinetics and performance of a polymeric annihilator. In comparison to a monomeric upconversion system that only functions using xTTA, we show that upconversion in a polymeric annihilator is efficient also at extremely low annihilator concentrations and that the overall kinetics is significantly faster. The presented results show that intramolecular photon upconversion is a versatile concept for the development of highly efficient solid-state photon upconversion materials.

A set $S \subseteq {{\mathbb {F}}_{2}^{n}}$ is called degree-d zero-sum if the sum ${\sum }_{s \in S} f(s)$ vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n − d − 1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream.

Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S 2 = 0 . We classify all maximal zero product subsets of Q by proving that the map R ↦ R ∩ LeftAnn ( R ) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.

Photon upconversion (PUC) via triplet–triplet annihilation (TTA) from near-infrared (NIR) to blue photons could have important applications especially to bioimaging and drug delivery accompanied by photochemical reaction. The fundamental challenges in achieving this has been the large anti-Stokes shift combined with the need to efficiently sensitize within the biological transparency window (700–900 nm). This calls for materials combinations with minimal energy losses during sensitization and minimal energy requirements to drive efficient TTA. Here, we demonstrate efficient PUC converting from NIR energy to blue photons using the commercially available material 9,10-bis[((triisopropyl)silyl)ethynyl]anthracene (TIPS-Ac) as the annihilator. With a conventional triplet sensitizing system, TIPS-Ac performed TTA with an efficiency of 77 ± 3% despite a relatively small driving force, compared to conventional TTA material converting from NIR to blue, for the TTA of less than 0.32 eV. Combined with Pt(II) meso-tetraphenyltetrabenzoporphine (PtTPBP), which is a heavy atom triplet sensitizer that directly generates triplets upon NIR photon excitation, the resulting system allowed for an anti-Stokes shift of 1.03 eV. Our results highlight the use of direct triplet generation via NIR excitation as a useful path to achieving large anti-Stokes shift and also show that high TTA efficiencies can be achieved even in the absence of large driving energies for the TTA process.

This paper is a continuation to \cite{Gazaki2017}. For every integer $n\geq 1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\xrightarrow{s_n} H^2(k,G_1[n]\otimes G_2[n])$, where $k$ is a finite extension of $\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and $K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mild assumptions, we describe the exact annihilator of the image of $s_n$ under the Tate duality perfect pairing, $H^2(k,G_1[n]\otimes G_2[n])\times H^0(k,Hom(G_1[n]\otimes G_2[n],\mu_n))\rightarrow\mathbb{Z}/n$. An important special case is when both $G_1, G_2$ are abelian varieties with split semistable reduction. In this case we prove a finiteness result, which gives an application to zero-cycles on abelian varieties and products of curves., Comment: 23 pages

The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R⋉M is an A-ring if and o...

We solve the following open problem in the negative: does the power series ring R [ [ x ] ] of a ring R inherit from R the property of every faithful right ideal being cofaithful? In other words, we construct a ring R such that every faithful right ideal of R is cofaithful, i.e., whenever the right annihilator rann R ( K ) of a right ideal K in R is {0}, then rann R ( Y ) = { 0 } for some non-empty finite subset Y of K, but R [ [ x ] ] does not have the mentioned property.

In this paper we construct a ring A which has annihilator condition (a.c.) and we show that neither A [ x ] nor A [ [ x ] ] has this property. This answers in negative a question asked by Hong, Kim, Lee and Nielsen. We also show that there is an algebra A which does not have annihilator condition while both A [ x ] and A [ [ x ] ] have this property.

Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.

We construct a new analogue of the BGG category O for the infinite-dimensional Lie algebras g = sl ( ∞ ) , o ( ∞ ) , sp ( ∞ ) . A main difference with the categories studied in [9] and [2] is that all objects of our category satisfy the large annihilator condition introduced in [5] . Despite the fact that the splitting Borel subalgebras b of g are not conjugate, one can eliminate the dependency on the choice of b and introduce a universal highest weight category OLA of g -modules, the letters LA coming from “large annihilator”. The subcategory of integrable objects in OLA is precisely the category T g studied in [5] . We investigate the structure of OLA , and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives. We also complete the annihilators in U ( g ) of simple objects of OLA .

We implement the notion of annihilator into fuzzy subgroups of an abelian group. There are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems. In this paper we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in Euclidean spaces. We find that in the natural algebraic duality of a group, a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator. We also study the behavior of annihilators with respect to unions and intersections. Some illustrative examples of annihilators of fuzzy subgroups are shown, both with finite and infinite rank.

Dike swarms are igneous structures of continental expression accounting for major episodes of magmatism in igneous provinces, mantle plume heads, and continental breakup. In regional magnetic maps, dike swarms are recognized by high-amplitude lineaments indicative of lengthy and juxtaposed magnetized bodies. High-anomaly amplitudes from such tabular (2D) bodies tend to obscure lower amplitude contributions from localized 3D sources, representative of magmatic structures that once served as magma plumbing and storage. The recognition of such subtle signals with conventional filtering techniques is prevented due to spectral overlapping of individual contributions. We have developed a processing scheme to remove contributions from elongated, homogeneous sources to make clear contributions from 3D sources located below, in the middle of, or above a framework of elongated homogeneous sources. The canceling of 2D fields is accomplished by evaluating the horizontal component of the magnetic anomaly along the lineament strike, which for true elongated and homogeneous sources gives a null response. The gradient intensity of the transformed field is then evaluated to enhance residual fields over 3D sources. Lineaments thus removed identify tabular bodies with homogeneous magnetization, interpreted as being indicative of the uniform distribution (mineral type, concentration, and grain-size distribution) of magnetic carrier content in the rock. We evaluated our technique with synthetic data from multiple 2D-3D interfering sources and then applied it to interpret airborne data from the Ponta Grossa Dike Swarm of the Paraná-Etendeka Magmatic Province in Southeastern Brazil.

We improve results and proofs of our earlier paper and show that the ideal in question is a complete intersection. Let $${\mathbb {K}}$$ be a field and $$\mathrm {M}={\mathbb {K}}[x^{-1},z^{-1}]$$ denote the $${\mathbb {K}}[x,z]$$ submodule of Macaulay’s inverse system $${\mathbb {K}}[[x^{-1},z^{-1}]]$$ . We regard $$z\in {\mathbb {K}}[x,z]$$ and $$z^{-1}\in \mathrm {M}$$ as homogenising variables. An inverse form $$F\in \mathrm {M}$$ has a homogeneous annihilator ideal $${\mathcal {I}}_F$$ . In our earlier paper we inductively constructed an ordered pair ( $$f_1$$ , $$f_2$$ ) of forms in $${\mathbb {K}}[x,z]$$ which generate $${\mathcal {I}}_F$$ . We give a significantly shorter proof that accumulating all forms for F in our construction yields a minimal grlex Groebner basis $${\mathcal {F}}$$ for $${\mathcal {I}}_F$$ (without using the theory of S polynomials or distinguishing three types of inverse forms) and we simplify the reduction of $${\mathcal {F}}$$ . The associated Groebner basis algorithm terminates by construction and is quadratic. Finally we show that $$f_1,f_2$$ is a maximal $${\mathbb {K}}[x,z]$$ regular sequence for $${\mathcal {I}}_F$$ , so that $${\mathcal {I}}_F$$ is a complete intersection.

The class of finite-dimensional unital right alternative superalgebras with semisimple strongly alternative even part over an algebraically closed field are studied. We introduce the concept of weak annihilator of a superalgebra and prove that it is a superideal, and the superalgebra itself splits into the direct sum of its weak annihilator and a superalgebra with a zero weak annihilator. We also classify superalgebras with zero weak annihilator in this class of superalgebras.

Let $G$ be a semiabelian variety defined over a field of characteristic $0$, endowed with an endomorphism $\Phi$. We prove there is no proper subvariety $Y\subset G$ which intersects the orbit of each periodic point of $G$ under the action of $\Phi$. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras., Comment: 23 pages

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

We classify all skew braces of Heisenberg type for a prime number p > 3 . Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew braces and Hopf–Galois theory, we can determine all Hopf–Galois structures of Heisenberg type on Galois field extensions of fields of degree p 3 .

We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteq Mat(m,n,R). The standard question of Singularity Theory is the finite-(\Sigma,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces to \Sigma and to the orbit, T_{(\Sigma,A)}, T_{(GA,A)} , and their quotient, the tangent module to the miniversal deformation. In particular, the order of determinacy is controlled by the annihilator of this tangent module. In this work we study this tangent module for the group action GL(m,R)\times GL(n,R) on Mat(m,n,R) and various natural subgroups of it. We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules etc., Comment: The final version

The order dual $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$ of the set $${{\rm Fil}{R_{R}}}$$ of all right topologizing filters on a fixed but arbitrary ring R is a complete lattice ordered monoid with respect to the (order dual) of inclusion and a monoid operation ‘ $${:}$$ ’ that is, in general, noncommutative. It is known that $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$ is always left residuated, meaning, for each pair $${\mathfrak{F}, \mathfrak{G} \in {\rm Fil}{R_{R}}}$$ there exists a smallest $${\mathfrak{H} \in {\rm Fil}{R_{R}}}$$ such that $${\mathfrak{H}: \mathfrak{G} \supseteq \mathfrak{F}}$$ , but is not, in general, right residuated (there exists a smallest $${\mathfrak {H}}$$ such that $${\mathfrak{G} : \mathfrak{H} \supseteq \mathfrak{F}}$$ ). Rings R for which $${[{\rm Fil}{R_{R}}]^{{\rm du}}}$$ is both left and right residuated are shown to satisfy the DCC on left annihilator ideals and possess only finitely many minimal prime ideals. It is shown that every maximal ideal P of a commutative ring R gives rise to an onto homomorphism of lattice ordered monoids $${\hat{\varphi}_{P}}$$ from $${[{\rm Fil}{R}]^{{\rm du}}}$$ to $${[{\rm Fil}{R_{P}}]^{{\rm du}}}$$ where RP denotes the localization of R at P. The kernel $${\equiv_{\hat{\varphi}_{P}}}$$ of $${\hat{\varphi}_{P}}$$ is a congruence on $${[{\rm Fil}{R}]^{{\rm du}}}$$ whose properties we explore. Defining $${{\rm Rad}({\rm Fil}{R})}$$ to be the intersection of all congruences $${\equiv_{\hat{\varphi}_{P}}}$$ as P ranges through all maximal ideals of R, we show that for commutative VNR rings R, $${{\rm Rad}({\rm Fil}{R})}$$ is trivial (the identity congruence) precisely if R is noetherian (and thus a finite product of fields). It is shown further that for arbitrary commutative rings R, $${{\rm Rad}({\rm Fil}{R})}$$ is trivial whenever $${{\rm Fil}{R}}$$ is commutative (meaning, the monoid operation ‘ $${:}$$ ’ on $${{\rm Fil}{R}}$$ is commutative). This yields, for such rings R, a subdirect embedding of $${[{\rm Fil}{R}]^{{\rm du}}}$$ into the product of all $${[{\rm Fil}{R_{P}}]^{{\rm du}}}$$ as P ranges through all maximal ideals of R. The theory developed is used to prove that a Prufer domain R for which $${{\rm Fil}{R}}$$ is commutative, is necessarily Dedekind.

Let R be a commutative Noetherian ring, M be a finitely generated R-module and n be a non-negative integer. In this article, it is shown that for a positive integer t, there is a finitely generated...

Let $$(R, {\mathfrak {m}}, k)$$ be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called $$\underline{\mathsf {h}}$$ -length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an $${\mathfrak {m}}$$ -primary cohomological annihilator, if there is a bound on the $$\underline{\mathsf {h}}$$ -length of all modules appearing in $$\mathsf {CM}$$ -support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. Namely, it is shown that R is of finite $$\mathsf {CM}$$ -type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule $$\omega $$ , as defined by Auslander and Reiten. It will turn out that, $$\omega $$ -Gorenstein projective modules with bounded $$\mathsf {CM}$$ -support are fully decomposable. In particular, a Cohen–Macaulay algebra $$\Lambda $$ is of finite $$\mathsf {CM}$$ -type if and only if every $$\omega $$ -Gorenstein projective module is of finite $$\mathsf {CM}$$ -type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules.

We study some “almost preserver” problems on von Neumann algebra modules. More precisely, we study (1) maps which “almost preserve” the right or left annihilator; (2) the “almost band preservers” – that is, maps which “almost preserve corners”, and (3) “almost centralizers,” which almost preserve module actions. Under certain conditions, we show that maps of these types are automatically continuous, and can be approximated by maps which precisely preserve these relations; often, the operators from the latter class are multiplication operators.

In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology and the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an - or -space all its homologically trivial modules possess the Dunford–Pettis property.

Let R be a commutative ring and let M be an R-module. For a ∈ R, AnnM(a) = {m ∈ M : am = 0} is said to be an annihilator submodule of M. In this paper, we study the property of being prime or essential for annihilator submodules of M. Also, we introduce the annihilator essential graph of equivalence classes of zero divisors of M, AER(M), which is constructed from classes of zero divisors, determined by annihilator submodules of M and distinct vertices [a] and [b] are adjacent whenever AnnM(a) + AnnM(b) is an essential submodule of M. Among other things, we determine when AER(M) is a connected graph, a star graph, or a complete graph. We compare the clique number of AER(M) and the cardinal of m −AssR(M).

Let $R$ be a ring, a mapping $F:R\rightarrow R$ together with a mapping $d:R\rightarrow R$is called a multiplicative (generalized)-reverse derivation if$F(xy)=F(y)x+yd(x)$ for all $x,y\in R$. The aim of this note is toinvestigate the commutativity of prime rings admittingmultiplicative (generalized)-reverse derivations. Precisely, it isproved that for some nonzero element $a$ in $R$ the conditions:$a(F(xy)\pm xy)=0$, $a(F(x)F(y)\pm xy)=0$, $a(F(xy)\pmF(y)F(x))=0$, $a(F(x)F(y)\pm yx)=0$, $a(F(xy)\pm yx)=0$ aresufficient for the commutativity of $R$. Moreover, we describe thepossible forms of generalized reverse derivations of prime rings.

Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $\mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.