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In this note we prove that the -points of the representation Artin-stack of n-dimensional representations of an affine ℂ-algebra R correspond to ℂ-algebra morphisms R → A n where A n is an Azumaya algebra of degree n over C. We connect this to the theory of D-branes and Azumaya noncommutative geometry, developed by Chien-Hao Liu and Shing-Tung Yau in a series of articles [9]–[14]. [ABSTRACT FROM PUBLISHER]

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ⊆ T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T. [ABSTRACT FROM PUBLISHER]

The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products of adequate semigroups by left regular bands. [ABSTRACT FROM PUBLISHER]

We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories. [ABSTRACT FROM PUBLISHER]

Recently in [10, 11], the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. In [5], a stronger version of state MV-algebras, called state-morphism MV-algebras, was given. In this article, we present Stone Duality Theorems for (i) the category of Boolean algebras with a fixed state-operator and the category of compact Hausdorff topological spaces with a fixed idempotent continuous function, and for (ii) the category of weakly divisible σ-complete state-morphism MV-algebras and the category of Bauer simplices whose set of extreme points is basically disconnected and with a fixed idempotent continuous function. [ABSTRACT FROM PUBLISHER]

Here we study the structure of a group with the presentation by conjugation which has been assigned to an extended affine root system R. We show that is isomorphic to the direct product of the extended affine Weyl group 𝒲 of R and the kernel of a natural epimorphism . The article also shows that has structural features that are well known for 𝒲. [ABSTRACT FROM PUBLISHER]

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding. [ABSTRACT FROM PUBLISHER]

We investigate the least studied class of differential rings—the class of differential rings of nonzero characteristic. We present the notion of differentially closed quasifield and develop geometrical theory of differential equations in nonzero characteristic. The notions of quasivariety and its morphisms are scrutinized. Presented machinery is a basis for reduction modulo p for differential equations. [ABSTRACT FROM PUBLISHER]

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary. [ABSTRACT FROM AUTHOR]

We introduce, in this article, a partial order on quasi-morphic rings and investigate the behavior of regular elements in such rings. Further, the cleanness of regular elements in quasi-morphic rings having internal-cancelltion has been derived. [ABSTRACT FROM AUTHOR]

We study here the nonzero composite of three irreducible morphisms between indecomposable modules lying in the fourth power of the radical. [ABSTRACT FROM AUTHOR]

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules. [ABSTRACT FROM AUTHOR]

Tom Dieck introduced a commutative triangle whereby the exponential morphism from the Burnside functor to the unit functor is factorized through the real representation functor. Tornehave introduced a p-adic variant of the exponential morphism. His construction involves real representations that are not well defined up to isomorphism. To obtain a well defined commutative triangle, we introduce the orientation functor, a quotient of the real representation functor. [ABSTRACT FROM AUTHOR]

Let H and H' be two ample line bundles over a smooth projective surface X, and M(H) (resp. M(H')) the coarse moduli scheme of H-semistable (resp. H'-semistable) sheaves of fixed type (r, c1, c2). We construct a sequence of blowing-ups which describes how M(H) differs from M(H') when r is arbitrary and the wall of fixed type separating H and H' is not necessarily good. Means we here utilize are elementary transforms and the notion of a sheaf with flag. [ABSTRACT FROM AUTHOR]

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack 0 consisting of nodal curves of genus 0. Here we define the basic classes: the classes of strata and the Mumford classes. [ABSTRACT FROM AUTHOR]

In this article, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors. [ABSTRACT FROM AUTHOR]

Banaschewski defines and gives sufficient conditions on a category A and a subclass M of its monomorphisms under which M-injectivity properly behaves with respect to the notions such as M-absolute retract, M-essentialness, and the existence of M-injective hulls. In this article, taking A to be the category of acts over a semigroup S and Md to be the class of sequentially dense monomorphisms, introduced and studied by Ebrahimi, Giuli, Mahmoudi, Moghaddasi, and Shahbaz, we study the behaviour of Md-injectivity (also called sequentially dense injectivity). We show, among other things, that S2 = S is a sufficient, but not necessary, condition for the behaviour of s-injectivity. Finally, we give an explicit description of the s-injective hulls of acts when they exist. [ABSTRACT FROM AUTHOR]

We study deformations of Leibniz algebra morphisms over a commutative local algebra base with 1. We construct the associated deformation cohomology that controls deformations using the cochain complex defining the Leibniz cohomology. [ABSTRACT FROM AUTHOR]

Let C be an arbitrary category. We study strongly involutory functors on C, defined as involutory contravariant endofunctors of C acting as identity on objects. Motivating examples can be constructed if we think at the transpose of a matrix, the adjoint of a linear continuous operator between two Hilbert spaces, and the inverse of a morphism in a groupoid. We show how a strongly involutory functor on a skeleton of C extends to C, and we apply this to find all such functors for a groupoid. We describe and classify up to a natural equivalence all strongly involutory functors on the category of finite dimensional vector spaces over a field. Strongly involutory functors with a special property related to generalized inverses of morphisms are studied. [ABSTRACT FROM AUTHOR]

In his previous article, the author has defined a higher version of the Nash blowup. In this article, we will introduce another higher version and prove that it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions. [ABSTRACT FROM AUTHOR]

A (commutative integral) domain is called a straight domain if A ↪ B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ↪ (I:KI) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that AP is a valuation domain, then the CPI-extension C(P) := A + PAP is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ↪ C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains. [ABSTRACT FROM AUTHOR]

For a class L of right modules we investigate the relations of monomorphic L-envelopes and epimorphic L-covers over pullback-pushout diagrams, and obtain some isomorphism equations between the envelopes of covers and the covers of envelopes. [ABSTRACT FROM AUTHOR]

We give a general geometrical procedure to construct nilpotent morphisms Φ : F → F(d), with F a vector bundle on 1, obtaining an analog of the Jordan canonical form. We investigate the possible splitting types of F in dependence on the degeneration behavior of Φ. Applications to nilpotent matrices with an arbitrary number of variables are also given. [ABSTRACT FROM AUTHOR]

Let X1, X2,..., Xn be realcompact spaces and Z be a topological space. Let π:C(X1) × C(X2) × ··· × C(Xn) → C(Z) be a Riesz n-morphism. We show that there exist functions σi : Z → Xi (i = 1, 2,..., n) and w ∈ C(Z) such that [image omitted] and σ1, σ2,..., σn are continuous on {z : w(z) ≠ 0}. This fact extends a result in Boulabiar (2002) and leads to one of the main results in Boulabiar (2004) with a more elementary proof. [ABSTRACT FROM AUTHOR]

Let C and D be two corings over a ring A and [image omitted] be a morphism of corings. We investigate the situation when the associated induced (“corestriction of scalars”) functor MC→ MD is a Frobenius functor, and call these morphisms Frobenius extensions of corings. The characterization theorem generalizes notions such as Frobenius corings and is applied to several situations; in particular, provided some (general enough) flatness conditions hold, the notion proves to be dual to that of Frobenius extensions of rings (algebras). Several finiteness theorems are given for each case we consider; these theorems extend existing results from Frobenius extensions of rings or from Frobenius corings, showing that a certain finiteness property almost always occur for many instances of Frobenius functors. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P0. We compare the posets TA and TB of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: TA → TB and e: TB → TA such that re = id. Moreover, e induces a full embedding of the quiver of TB into that of TA, whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second. [ABSTRACT FROM AUTHOR]

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding uA:A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (uA, uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are "u-extendable" is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ∀ B(frA(A, B) = U (A, B)), and, ∀ B ≠ 0(frA(B, A) = U(B, A)). [ABSTRACT FROM AUTHOR]

We find a nonreflexive projective 3-fold in P5 with birational Gauss map onto its image, in any positive characteristic. This gives a negative answer, also in characteristic p > 2, of the following Kleiman-Piene's question: Does the separability of the Gauss map imply reflexivity? Kaji already gave a negative answer in characteristic 2. [ABSTRACT FROM AUTHOR]

An algebraic deformation theory of morphisms of dual Leibniz algebras is obtained. [ABSTRACT FROM AUTHOR]

We investigate adjoint and Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory. [ABSTRACT FROM AUTHOR]

Tue study of coset monoids was initiated by Schein in 1966. The coset monoid of a group G, denoted ...(G), consists of all cosets of all subgroups of G. We show how to generalize ...(G) by constructing a monoid ...ℒ(G) of cosets of subgroups from a semilattice (ℒ, ∨ℒ) of subgroups of G which satisfies certain conditions. When (ℒ, ∨ℒ) = (...(G), ∨) is the join semilattice of all subgroups of G , we recover the coset monoid ...(G). The class of semigroups which are isomorphic to some ...ℒ(G) has a very simple description. Elements of the monoids in this class may be represented by cosets of subgroups of their group of units. We show that the factorizable part of the symmetric (resp. dual symmetric) inverse semigroup does not (resp. does) belong to this class. [ABSTRACT FROM AUTHOR]

Let S be a partially ordered monoid, or briefly, pomonoid. A right S-poset (often denoted AS) is a poset A together with a right S-action (a, s) as that is monotone in both arguments and that satisfies the conditions a(st) = (as)t and a1 = 1 for all a ∈ A, s, t ∈ S. Left S-posets SB are defined analogously, and the left or right S-posets form categories, S-POS and POS-S, whose morphisms are the monotone maps that preserve the S-action. In these categories, as in the category POS of posets, the monomorphisms and epimorphisms are the injective and surjective morphisms, respectively, but the embeddings and quotient maps have stronger properties; in particular, an embedding is a monomorphism that is also an order embedding. A tensor product AS ⊗ SB exists (a poset) that has the customary universal property with respect to balanced, bimonotone maps from A × B into posets. Various flatness properties of AS can be defined in terms of the functor AS ⊗− from S-POS into POS. More specifically, an S-poset AS is called flat if the induced morphism AS ⊗ SB → AS ⊗ SC is injective whenever SB → SC is an embedding in S-POS this means that, for all SB and all a, a' ∈ A and b, b' ∈ B, if a ⊗ b = a' ⊗ b' in A ⊗ B, then the same equality holds in A ⊗ (Sb ∪ Sb'). AS is called (principally) weakly flat if the induced morphism above is injective for all embeddings of (principal) left ideals into SS. Similarly, AS is called po-flat if the functor AS ⊗− preserves embeddings: for this definition, replace = by ≤ in the description above (see Shi, 2005). Weak and principally weak versions of po-flatness are defined in an obvious way. In the present article, we first consider flatness properties for one-element and Rees factor S-posets. We present examples that distinguish between various types of flatness and the corresponding, generally stronger, notions of po-flatness. We then initiate a study of absolute flatness for pomonoids: a monoid (respective pomonoid) S is called right absolutely flat if all right S-acts (respective S-posets) are flat. The findings for absolute flatness of pomonoids are markedly different from the corresponding unordered results. [ABSTRACT FROM AUTHOR]

In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five. [ABSTRACT FROM AUTHOR]

If M and N are modules, the concept of semiregularity (and regularity) of hom ( M , N ) is defined and studied, and the connection with the relative direct injective- and direct projective-properties is established. The relationship of semiregularity to the Jacobson radical of hom ( M , N ), to the singular and cosingular ideals of hom ( M , N ), and to the notion of lying over or under a direct summand, is described, and the basic results in the module case are extended. Communicated by R. Wisbauer. [ABSTRACT FROM AUTHOR]

Let D be a Prüfer domain, and denote by ± b ℑ( D ) the multiplicative group of all invertible fractional ideals of D , ordered by A ≤ B if and only if A ⊇ B . Denote by G i the value group of the valuation associated with the valuation ring D M i , where { M i } i ∈ I is the collection of all maximal ideals of D . In this note we prove that the natural map from ± b ℑ( D ) into ± b ∏ i ∈ I G i is an isomorphism onto the cardinal sum ± b ∐ i ∈ I G i if and only if D is h -local. As a corollary, the group of divisibility of an h -local Bézout domain is isomorphic to ± b ∐ i ∈ I G i , the notation being as above. [ABSTRACT FROM AUTHOR]

After the proof of a not very known refinement of the Noether Normalization Theorem, we obtain two sharp degree bounds for the geometric degree of a dominating morphism of irreducible affine algebraic varieties and for the degree of the components of the inverse of an isomorphism of such varieties, generalizing by this way the well-known Gabber bound for automorphisms of affine spaces. [ABSTRACT FROM AUTHOR]

In this note,we answer a question of Hong et al. (2003) by proving that if a is a monomorphism of a reduced ring R , and R is a-skew Armendariz, then R is a-rigid. [ABSTRACT FROM AUTHOR]

LetΛ be an artin algebra over a local commutative artinian ring k. We consider the category P(Λ) whose objects are morphims f : P → Q with P and Q projective leftΛ-modules. In P(Λ), we introduce an exact structure in the sense of Gabriel and Roiter [Gabriel, P., Roiter, A. V. (1992). Representations of finite-dimensional algebras. In: Kostrikin, A. I., Shafarevich, I. V. eds. Encyclopaedia of the Mathematical Sciences. Vol. 73. Springer. Algebra VIII.] or equivalently Quillen [Quillen, D. (1973). Higher Algebraic K-Theory, 1. SLNM 341. Berlin: Springer, pp. 85–147], then we describe the corresponding projectives and injectives. For p(Λ), the full subcategory whose objects are morphisms f : P → Q with P and Q finitely generated, we prove a relation between the Hom-functor and the Ext-functor. From here, we can prove the existence of almost split sequences in p(Λ) and its relation with the almost split sequences for leftΛ-modules. [ABSTRACT FROM AUTHOR]

In any finitely complete category, there is an internal notion of normal monomorphism. We give elementary conditions guaranteeing that a normal section s: Y → X of an arrow f: X → Y produces a direct product decomposition of the form X ≃ Y × W. We then show how these conditions gradually vanish in various algebraic contexts, such as Maltsev, protomodular and additive categories. [ABSTRACT FROM AUTHOR]

Homological algebra based on fp-injective and fp-flat modules is studied. It can be realized as relative homological algebra corresponding to the proper class of monomorphisms Also, the corresponding homological functors Extf and Torf as well as various homological dimensions are investigated. [ABSTRACT FROM AUTHOR]

In this paper we examine isomorphism classes of algebras which are generalizations of the enveloping algebra of the Heisenberg Lie algebra. [ABSTRACT FROM AUTHOR]

We determine the number of elements of order two in the group of normalized units V(F2G) of the group algebra F2G of a 2-group of maximal class over the field F2 of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(F2G) and V(F2H) are isomorphic if and only if G and H are isomorphic. [ABSTRACT FROM AUTHOR]

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto φ(C) ⊂ Pr. We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H0(C,L²) can be written as α² + β² + γ², with α, β, γ ∈ H0 (C, L). If there are no quadrics of rank 3 containing φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem. [ABSTRACT FROM AUTHOR]

Let A be a basic finite dimensional k-algebra which contains a semisimple algebra S separable over k with &Lambda = S ⊕ J, where J is the Jacobson radical of Λ. The category P(Λ) of radical morphisms between projective left Λ-modules has an exact structure. For φ1,φ2 in P(Λ), denote by ExPtφ1,φ2 the corresponding extension group, A finitely generated left Λ-module M has rigid minimal projective presentation Pφ→ Q → M → 0 if ExtP(φ,φ)= 0. We will see that if M is indecomposable and has rigid minimal projective presentation then EndΛ(M)/radEndΛ((M) ...EndΛ(T) for T a simple Λ-module. [ABSTRACT FROM AUTHOR]

We prove that for a smooth projective variety X of dimension d defined over a finite field k, the structure map σ : X [xrarr] Spec k induces an isomorphism σ∗ : CHd+1(X, 1) ≅ CH1(k, 1) = k*. We also prove that the higher Chow groups CHd+s−1(X, s) of one-dimensional cycles are torsion for s ≥ 2. [ABSTRACT FROM AUTHOR]

Let (A, B, f) and (U, V, g) be C-algebras and ψ : (A, B, f) → (U, V, g) a C-algebra homomorphism. Let (A, Bˆf, f) and (U, Vˆg, g) be the dual C-algebras of (A, B, f) and (U, V, g), respectively. In this article we define a dual morphism ψˆ : (U, Vˆg, g) →  (A, Bˆf, f) for ψ, and use this dual morphism to establish the bijection between the quotient subsets of B and the C-subsets of its dual Bˆf obtained in [Blau, H. I. (1995a). Quotient Structures in C-algebras. J. Algebra 177:297–337]. This approach provides a conceptual proof of this very important bijection. [ABSTRACT FROM AUTHOR]

The elementary divisors of the Gram matrices of Specht modules S[SUPλ] over the symmetric group are determined for two-row partitions and for two-column partitions λ. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition λ of n at a prime p > n - λ[SUB1], the only possible non trivial composition factor of S[SUPλ,SUBF[SUBp]] is induced by the morphism of Carter and Payne, as shown by means of Kleshchev's modular branching rule. This enables the Jantzen filtration to be calculated in this case as well. [ABSTRACT FROM AUTHOR]

Given a contravariant functor F : [Cscr] → [Sscr]ets for some category [Cscr], we say that F ([Cscr]) (or F) is generated by a pair (X, x) where X is an object of [Cscr] and x ∈ F(X) if for any object Y of [Cscr] and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor [Escr]xt (-, M) : [Escr] → [Sscr]ets, where M is a left R-module and [Escr] is the class (considered as full subcategory of left R-modules) of injective left R-modules. [ABSTRACT FROM AUTHOR]

Studies the perpendicular categories of exceptional vector bundles on wild weighted projective lines. Use of algebraically closed field k for derivations; Description of morphism between vector bundles.

In the case when R is a Dedekind domain of positive prime characteristic p > 0, we give a parameterisation of the isomorphism classes of the injective indecomposable R∞-modules. Here R∞ is the smallest ring extension of R on which the Frobenius morphism acts bijectively. Further, in this case, every injective R∞-module is the injective hull of a direct sum of injective indecomposable R∞-modules. [ABSTRACT FROM AUTHOR]