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We explore two modules F and K over a subalgebra U of the quantum group U q (g l (∞)) . The first U -module is the Fock space related to the natural module over the quantum group. If we let O + to be the category of finite-dimensional o s p (2 | 2) -modules of integral weights, the second U -module K is the vector space obtained by tensoring the Grothendieck group K (O +) by the field Q . Using thick tensor ideals and a U -isomorphism, we compute the socles of F and K . [ABSTRACT FROM AUTHOR]

The concept of the spectral rank of an element in a ring is defined, and it is shown to be a genuine generalization of the same concept first studied in the setting of a Banach algebra. Furthermore, we prove that many of the desirable properties of this rank are still valid in the more abstract setting and give several examples to support and motivate the given definition. In particular, we are able to show that a nonzero idempotent of a semiprimitive and additively torsion-free ring is minimal if and only if it has a spectral rank of one. We also discover a precise connection between the spectral rank of an element in a ring and a purely algebraic definition of rank considered only recently by N. Stopar in [Rank of elements of general rings in connection with unit-regularity, J. Pure Appl. Algebra 224 (2020) 106211]. Specifically, we are able to show that if an element a has a finite algebraic rank in a semiprimitive and additively torsion-free ring, then a has the exact same spectral rank. An extra condition under which the converse holds true is also provided, and connections to the socle are identified. Finally, for both of these extended notions of rank considered in the setting of a ring, we prove a generalized Frobenius Inequality. [ABSTRACT FROM AUTHOR]

AbstractLet F be a field. We study the structure of finite groups G for which the socle of the center of the group algebra FG is an ideal in FG. In previous work (Brenner-Külshammer, 2024), we studied which general implications this ring-theoretic property of FG has on the structure of G. In the present article, we now focus on a detailed structural analysis of the groups with this property. We showcase its strength by deriving a complete classification for a prototype class of groups. [ABSTRACT FROM AUTHOR]

There has been a growing research interest in exploring rings whose modules are either injective or far from being injective to a specified degree. As is commonly observed in each of such scenarios, rings can be broken as a direct sum of a semisimple Artinian ring and an indecomposable ring having zero or (essential) homogeneous right socle. Although this phenomenon has become increasingly expected and less surprising with every new study, proving it has never become easier, and in most cases, it has been obtained as a result of considerable efforts. This work aims to bring such studies to a common denominator and tries to explain why this phenomenon occurs each time we deal with the same type of problem. [ABSTRACT FROM AUTHOR]

We will give an algebraic characterization of generalized Fredholm operators in terms of projections by means of Schmoeger (Demonstr Math XXXII(3):595–604, 1999, Therorem 1.1). More precisely, for T ∈ B (X) which is the Banach algebra of all bounded linear operators on a Banach space X , we shall prove the following fact: T is a generalized Fredholm operator if and only if there exists a projection P which commutes with T such that, TP is a Fredholm element in P B (X) P with jump j P B (X) P (T P) = 0 and (I d X - P) T is a nilpotent element in s o c ((I d X - P) B (X) (I d X - P)) . This characterization can open the way for us to extend the above Theorem to the more general context like that of a semisimple complex Banach algebras. [ABSTRACT FROM AUTHOR]

Let sl (2) ⋉ h n , n ≥ 1 , be the Galilean Lie algebra over a field of characteristic zero, here h n is the Heisenberg Lie algebra of dimension 2 n + 1 , and sl (2) acts on h n so that, sl (2) -modules, h n ≃ V (2 n - 1) ⊕ V (0) (here V(k) denotes the irreducible sl (2) -module of highest weight k). In this paper, we study the tensor product of two uniserial representations of sl (2) ⋉ h n . We obtain the sl (2) -module structure of the socle of V ⊗ W and we describe the space of intertwining operators Hom sl (2) ⋉ h n (V , W) , where V and W are uniserial representations of sl (2) ⋉ h n . The structure of the radical of V ⊗ W follows from that of the socle of V ∗ ⊗ W ∗ . The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for sl (2) ⋉ h n reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial (sl (2) ⋉ h n) -modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra sl (2) ⋉ a m where a m is the abelian Lie algebra of dimension m + 1 and sl (2) acts so that a m ≃ V (m) as sl (2) -modules. [ABSTRACT FROM AUTHOR]

In this paper, we define a condition termed as cohesiveness on a submodule N of the socle of QTAG-module M, which is necessary for N to be the socle of an isotype submodule of M. It is proved here that for the countably generated submodule N, this condition is both necesaary and sufficient. We further define a restricted condition called as completely cohesiveness and shows that this condition is necessary and sufficient for a subsocle T of a QTAG-module which supports a balanced submodule of M. [ABSTRACT FROM AUTHOR]

Let k be a field with characteristic zero, R be the ring $k[x_1, \cdots, x_n]$ k [ x 1 , ⋯ , x n ] and I be a monomial ideal of R. We study the Artinian local algebra $R/I$ R / I when considered as an R-module M. We show that the largest reduced submodule of M coincides with both the socle of M and the k-submodule of M generated by all outside corner elements of the Young diagram associated with M. Interpretations of different reduced modules is given in terms of Macaulay inverse systems. It is further shown that these reduced submodules are examples of modules in a torsion-torsionfree class, together with their duals; coreduced modules, exhibit symmetries in regard to Matlis duality and torsion theories. Lastly, we show that any R-module M of the kind described here satisfies the radical formula. [ABSTRACT FROM AUTHOR]

In this work, amply socle supplemented modules are defined and some properties of these modules are investigated. We prove that every π-projective and socle supplemented module is amply socle supplemented. We also prove that every factor module and every homomorphic image of an amply socle supplemented module are amply socle supplemented. Let M be a projective and socle supplemented R-module. Then every finitely M-generated R-module is amply socle supplemented. [ABSTRACT FROM AUTHOR]

Let g m = sl (2) ⋉ V (m) , m ≥ 1 , where V (m) is the irreducible sl (2) -module of dimension m + 1 viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial g m -modules consist of a family, say of type Z , containing modules of arbitrary composition length, and some exceptional modules with composition length ≤4. Let V and W be two uniserial g m -modules of type Z. In this paper we obtain the sl (2) -module decomposition of soc (V ⊗ W) by giving explicitly the highest weight vectors. It turns out that soc (V ⊗ W) is multiplicity free. Roughly speaking, soc (V ⊗ W) = soc (V) ⊗ soc (W) in half of the cases, and in these cases we obtain the full socle series of V ⊗ W by proving that soc t + 1 (V ⊗ W) = ∑ i = 0 t soc i + 1 (V) ⊗ soc t + 1 − i (W) for all t ≥ 0. As applications of these results, we obtain for which V and W , the space of g m -module homomorphisms Hom g m (V , W) is not zero, in which case is 1-dimensional. Finally we prove, for m ≠ 2 , that if U is the tensor product of two uniserial g m -modules of type Z , then the factors are determined by U. We provide a procedure to identify the factors from U. [ABSTRACT FROM AUTHOR]

Let G be a finite p -group, and α an automorphism of the group algebra F p G. Then α fixes the socle of F p G pointwise. More generally, if k is a field of characteristic p , and α is a k -algebra automorphism of kG , then α induces a linear action on the dimension subquotients of the group, and the action on the socle is scalar multiplication by the (p − 1) st power of the product of the determinants of this action. The scalar is thus an element of (k ×) p − 1. [ABSTRACT FROM AUTHOR]

Given a simple graph, consider the polynomial ring with coefficients in a field and variables identified with the edges of the graph. Given a non-empty even cardinality Eulerian subgraph and a choice of half of its edges, consider the homogeneous binomial obtained by taking the product of these edges minus the product of the remaining edges of the subgraph. We define a homogeneous ideal by taking as generators all binomials obtained in this way, varying the Eulerian subgraph and the choice of half of its edges, together with the squares of the variables of the ring. This ideal is related to the Eulerian ideal, introduced by Neves, Vaz Pinto and Villarreal. We call the corresponding quotient the Eulerian Artinian algebra associated with the graph. The goal of the present work is to study the socle of these algebras through the lens of graph theory. Our main results include a combinatorial characterization of a monomial basis of the socle, a characterization of Gorenstein Eulerian Artinian algebras in the case of bipartite graphs and the computation of the h-vector and socle degrees in the cases of a complete graph and a complete bipartite graph. [ABSTRACT FROM AUTHOR]

Dans cet article, nous décrivons le mariage chez les congolais en général et, chez les Yansi en particulier qui est considéré comme un moyen de perpétuer la lignée de l’ancêtre fondateur. Le mariage a un caractère communautaire, c’est-à-dire, il ne concerne pas que les deux conjoints, mais aussi l’implication totale de tous les membres des familles des concernés (le jeune garçon et la jeune fille). Il s’agit des pères, des mères, frères et sœurs des familles restreintes, élargies, mais aussi des communautés environnantes des mariés. Le mariage préférentiel trouve sa légitimité lorsque la dot a été versée lors du mariage de la grand-mère maternelle de la fille, la future épouse. Ce mariage est basé sur le consentement mutuel des conjoints, sans exclure l’apport des membres de deux familles de futurs mariés. Tout se base autour de l’amour, de l’acceptation mutuelle de l’un(e) envers l’autre. Le mariage préférentiel dans les milieux urbains a évolué dans sa forme et dans son fond. Ce qui se fait en ville est pareil de ce qui est fait et exigé en milieu rural. Le grand-père maternel (NGA-KIEY) a droit à la dot de sa petite-fille maternelle appelée « ETUIL » ou « KITUIL ». Il est bénéficiaire des biens en espèce et en nature de la dot de la mariée (épouse), sa petite-fille.

We study symmetric algebras A over a field F in which the Jacobson radical of the center of A, the socle of the center of A or the Reynolds ideal of A are ideals. [ABSTRACT FROM AUTHOR]

Let A and B be complex unital Banach algebras, and let σ (x) and σ ^ (x) denote the spectrum of x and its polynomially convex hull, respectively. A spectrally additive map ϕ : A → B is a surjective function which satisfies σ (x + y) = σ (ϕ (x) + ϕ (y)) for each x , y ∈ A . By establishing a new additive characterization of rank one elements in a Banach algebra, we prove that any spectrally additive map acting on a semisimple domain preserves rank one elements in both directions. This settles an open question raised in [1], which ultimately then classifies spectrally additive maps on a large class of Banach algebras as Jordan-isomorphisms. By refining the techniques in [1] even further, we are able to prove the following more general result: If A is semisimple and either A or B has an essential socle, then any surjective map ϕ : A → B with the property that σ ^ (x + y) = σ ^ (ϕ (x) + ϕ (y)) for each x , y ∈ A is a continuous Jordan-isomorphism. [ABSTRACT FROM AUTHOR]

Let A{\mathcal{A}} be a unital primitive C∗\ast -algebra. This article studies the properties of the B-Fredholm elements, the B-Weyl elements and the B-Browder elements in A{\mathcal{A}}. Particularly, this article describes the B-Fredholm element as the sum of a Fredholm element and a nilpotent element. In addition, the socle of A{\mathcal{A}} is characterized by the B-Fredholm elements.

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]

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda)$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu),\Delta (\lambda))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$. [ABSTRACT FROM AUTHOR]

We shall show a spectral mapping property of the generalized Fredholm spectrum in the more general context of Banach algebras. This is an extension of (Schmoeger in Demonstr Math XXXI(4):723–733, 1998, Theorem 3.3). More precisely, for a ∈ A which is a complex semisimple Banach algebra with identity, we shall prove that if g is a analytic function on a neighbourhood of the spectrum of a and if it has no zero in the generalized Fredholm spectrum of a then g(a) is a generalized Fredholm element in A . [ABSTRACT FROM AUTHOR]

We consider a bounded lattice (L, Λ, V) with the smallest element 0 and the greatest element 1. In this paper, we deal with the essentiality concepts associated with a lattice. For an arbitrary element ϑ of L, we define a ϑ-e-irreducible element in L, which is an analogy to the concept of the e-irreducible submodule in a module over a ring. It is well known that e-irreducible submodules have no proper essential extension. Indeed, we prove this remains true for elements in a bounded lattice. We establish a relation between the ϑ-complement and ϑ-e-irreducible element with suitable examples. We define the notion ϑ-socle and prove several properties when a lattice is compactly generated. Further, we construct a generalized complement graph of a distributive lattice and relate the properties such as connectedness, diameter, and cut vertices to atoms in a lattice. [ABSTRACT FROM AUTHOR]

Let A be a unital primitive C*-algebra. This paper studies the spectral theories of B-Weyl elements and B-Browder elements in A, including the spectral mapping theorem and a characterization of B-Weyl spectrum. In addition, we characterize the generalized Weyl's theorem and the generalized Browder's theorem for an element a A and f(a), where f is a complex-valued function analytic on a neighborhood of s(a). What's more, the perturbations of the generalized Weyl's theorem under the socle of A and quasinilpotent element are illustrated. [ABSTRACT FROM AUTHOR]

Let A be a unital semisimple Banach algebra. Denote the set of the generalized Fredholm elements in A by Φ g (A) . In this paper, we study the perturbations of the generalized Fredholm elements and the spectral mapping theorem of the generalized Fredholm spectrum. Furthermore, for a ∈ Φ g (A) , the conditions that f(a) is also a generalized Fredholm element are investigated, where f is a complex-valued function analytic on a neighborhood of σ (a) . In addition, the topological structure of Φ g (A) are discussed. As an application, the socle of a primitive C ∗ -algebra is characterized by the generalized Fredholm elements. [ABSTRACT FROM AUTHOR]

Let L be a zero-dimensional frame and ZL be the ring of integer-valued continuous functions on L. We associate with each sublocale of zL, the Banaschewski compactification of L, an ideal of ZL, and study the behaviour of these types of ideals. The socle of ZL is shown to be always the zero ideal, in contrast with the fact that the socle of the ring RL of real-valued continuous functions of L is not necessarily the zero ideal. The ring ZL has been shown by B. Banaschewski to be (isomorphic to) a subring of RL, so that ideals of the larger ring can be contracted to the smaller one. We showthat the contraction of the socle of RL to ZL is the ideal of ZL associated with the join (in the coframe of sublocales of zL) of all nowhere dense sublocales of zL. It also appears in other guises. [ABSTRACT FROM AUTHOR]

In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things, it is shown that there exist no rigid zero-dimensional Gorenstein singularities and rigid almost complete intersections. In the proof of the first result we exploit the canonical duality between homology and cohomology of the cotangent complex, while in the proof of the second we use a new method which is based on the properties of the torsion functor. In addition, we obtain highly efficient estimates for the dimension of the spaces of the first lower and upper cotangent functors of arbitrary zero-dimensional singularities, including the space of derivations. We also consider examples of nonsmoothable zero-dimensional noncomplete intersections and discuss some properties and methods for constructing such singularities using the theory of modular deformations, as well as a number of other applications. [ABSTRACT FROM AUTHOR]

Let $\mathcal{A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal{A}))$ the kernel of the hull of the socle of $\mathcal{A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal{A}$. This allows us to show that a "result" from Riesz theory in commutative Banach algebras is not true.

Let p and q be distinct odd primes. Let Γ = (V (Γ),E(Γ)) be a non-Cayley vertex-transitive graph of order pq. Let G ≤ Aut(Γ) acts primitively on the vertex set V (Γ). In this paper, we show that G is uniprimitive which is primitive but not 2-transitive and we obtain some information about p,q and the minimality of the Socle T = Soc(G)

There are three famous results in the field of rings of real valued continuous functions namely, M. H. Stone Theorem, Urysohn’s Extension Theorem and Gelfand Kolmogoroff Theorem. We study analogous results for rings of functions satisfying the property mentioned in the tittle of this article. [ABSTRACT FROM AUTHOR]

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]

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$ , then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$ -subspace lattice algebras, is a derivation. [ABSTRACT FROM AUTHOR]

Let A be a commutative complex semisimple Banach algebra. In this paper we continue the study of kh(soc(A)). Thus we will give, among other things, some new characterizations of this ideal in terms of the closure, in the spectral radius norm, of the socle of A. [ABSTRACT FROM AUTHOR]

Lifting homomorphism from modulesto modules or even from certain submodule to the modules have been importantboth in ring and module theory. In this note we study rings and modules whosesocles are relative ejective. Moreover we reduce our consideration to rings andmodules with injective socles which provides the dual notion to PSmodules.

Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring Cs(X) of all real valued clopen continuous functions on a topological space X. It is shown that every ƒ ∈ Cs(X) is constant on each quasi-component in X and using this fact we show that Cs(X) ≅ C(Y), where Y is a zero-dimensional s-quotient space of X. Whenever X is locally connected, we observe that Cs(X) ≅ C(Y), where Y is a discrete space. Maximal ideals of Cs(X) are characterized in terms of quasi-components in X and it turns out that X is mildly compact(every clopen cover has a finite subcover) if and only if every maximal ideal of Cs(X)is fixed. It is shown that the socle of Cs(X) is an essential ideal if and only if the union of all open quasi-components in X is s-dense. Finally the counterparts of some familiar spaces, such as Ps-spaces, almost Ps-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring Cs(X).

We prove that the tensor product of a simple and a finite dimensional sl n -module has finite type socle. This is applied to reduce classification of simple q (n) -supermodules to that of simple sl n -modules. Rough structure of simple q (n) -supermodules, considered as sl n -modules, is described in terms of the combinatorics of category O . [ABSTRACT FROM AUTHOR]

If R ⊆ S is an extension of commutative rings, we consider the lattice ([R, S], ⊆) of all the R-subalgebras of S. We assume that the poset [R, S] is both Artinian and Noetherian; that is, R ⊆ S is an FCP extension. The Loewy series of such lattices are studied. Most of main results are gotten in case these posets are distributive, which occurs for integrally closed extensions. In general, the situation is much more complicated. We give a discussion for finite field extensions. [ABSTRACT FROM AUTHOR]

Let A be a commutative complex semisimple Banach algebra. Denote by kh(soc(A)) the kernel of the hull of the socle of A. In this work we give some new characterizations of this ideal in terms of minimal idempotents in A. This allows us to show that a "result" from Riesz theory in commutative Banach algebras is not true. [ABSTRACT FROM AUTHOR]

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the n th (right) socle Z S n (A) : = Z (A) ∩ Soc n (A) of a finite-dimensional algebra A is a Morita invariant; this is a generalization of important Morita invariants — the center Z (A) and the Reynolds ideal Z S 1 (A). As an example, we also studied Z S n (F G) for the group algebra FG of a finite p -group G over a field F of positive characteristic p. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of Z S n (F G). In fact, such elements form a basis of Z S n (F G) for every integer 1 ≤ n ≤ p if G is powerful. As a corollary we have Soc p (F G) ⊆ Z (F G) if G is powerful. [ABSTRACT FROM AUTHOR]

For a finite subset M ⊂ [x1,..., xd] of monomials, we describe how to constructively obtain a monomial ideal I ⊆ R = K [ x 1 , ... , x d ] such that the set of monomials in Soc(I) ∖ I is precisely M, or such that M ¯ ⊆ R / I is a K-basis for the the socle of R/I. For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely the lattice structure of the monoid [x1,..., xd]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice (ℤ d , ≼) . Finally, we define and analyze zero-dimensional monomial ideals of R of type k, where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in ℤ d . [ABSTRACT FROM AUTHOR]

This paper studies the distribution of socle degrees of R/I[pe] when e is large, for a homogeneous ideal I in a two-dimensional standard-graded normal domain R in positive characteristic p. We prove that the distribution is very much related to the asymptotic slopes of the syzygy bundle Syz(I), which have been known to determine the Hilbert-Kunz multiplicity of I . [ABSTRACT FROM AUTHOR]

We identify the largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied. [ABSTRACT FROM AUTHOR]

For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms. Also it is shown that not only $ mathcal{F}_{mathcal P}L$ has Property (A) but also if $L$ has a finite number of atoms then the residue class ring $ mathcal{F}_{mathcal P}L/mathrm{Soc}( mathcal{F}_{mathcal P}L)$ has Property (A).

The set Fil R R of all right topologizing filters on a fixed but arbitrary ring R admits a monoid operation ' : ' that is in general noncommutative, even in cases where the ring R is commutative. Earlier results show (see [8]) that commutativity of the monoid operation ' : ', when imposed as a condition on Fil R R , manifests as a type of finiteness condition on R. In a quite separate and much earlier study, Shores [6] has shown that if R is a commutative semiartinian ring, then R will be artinian precisely if the first two terms in the Loewy series for RR, namely soc (R R) and soc 2 (R R) , are finitely generated. Shores goes further to produce examples which show that the finiteness of just soc (R R) exercises no constraint whatsoever on the length of RR. The main result of this paper asserts that a commutative semiartinian ring R will be artinian precisely if soc (R R) is finitely generated and the monoid operation ' : ' on Fil R R is commutative. A family of commutative semiartinian rings of Loewy length 3 is constructed and this used to delineate earlier theory. In particular, and within this family, rings R are exhibited such that (1) soc (R R) and soc 2 (R R) / soc (R R) have infinite length, yet (2) the monoid operation ' : ' on Fil R R is commutative. [ABSTRACT FROM AUTHOR]

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on. Further let contain precisely those functions f in C(X) for which for each ϵ > 0, {x ∈ X: |f (x)| ≥ ϵ} is a member of. Let stand for the set of all those points p in βX at which the stone extension f∗ for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally- , becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice ≡ the ideal of compact sets in X. Finally we observe that the ideals of the form of C(X) are no other than the z◦-ideals of C(X). [ABSTRACT FROM AUTHOR]

We study algebraic properties of powers of squarefree principal Borel ideals I, and show that astab(I) = dstab(I). Furthermore, the behaviour of the depth function depth S/Ik is considered. [ABSTRACT FROM AUTHOR]

Abstract The classification of all block-transitive 2 − (v , k , 1) designs has been known as a long-standing and still open problem. This work is a contribution to the general program to classify the pairs (D , G) in which D is a 2 − (v , k , 1) design and G act as a point-primitive block-transitive automorphism group of D with a Suzuki group Sz (q) socle. We determine the point-stabilizer G α ∩ Sz (q) and the block-stabilizer G B ∩ Sz (q), and the parameter of D is given. [ABSTRACT FROM AUTHOR]

In a recent paper, De Stefani and Núñez-Betancourt proved that for a standard-graded F-pure κ-algebra R, its diagonal F-threshold c(R) is always at least -α(R), where α(R) is the a-invariant. In this paper, we establish a refinement of this result in the setting of complete intersection rings. [ABSTRACT FROM AUTHOR]

It is shown that in some non-discrete topological spaces, discrete subspaces with certain cardinality are C-embedded. In particular, this generalizes the well-known fact that every countable subset of P-spaces are C-embedded. In the presence of the measurable cardinals, we observe that if X is a discrete space then every subspace of X (i.e., the Hewitt realcompactification of X) whose cardinal is nonmeasurable, is a C-embedded, discrete realcompact subspace of X. This generalizes the well-known fact that the discrete spaces with nonmeasurable cardinal are realcompact. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we will give a characterization of the socle of a semisimple Banach algebra by perturbation of full spectrum. That is, an element, saying x , lies in the socle if and only if for every element in the Banach algebra, the change under its perturbation (plus x) of full spectrum is finite. [ABSTRACT FROM AUTHOR]