Search

For a smooth affine algebra R of dimension d ≥ 3 over a field k and an invertible alternating matrix χ of rank 2 n , the group S p (χ) of invertible matrices of rank 2 n over R which are symplectic with respect to χ acts on the right on the set U m 2 n (R) of unimodular rows of length 2 n over R. In this paper, we prove that S p (χ) acts transitively on U m 2 n (R) if k is algebraically closed, d ! ∈ k × and 2 n ≥ d. [ABSTRACT FROM AUTHOR]

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper. [ABSTRACT FROM AUTHOR]

We prove that a ring R has a module M whose domain of projectivity consists of only some injective modules if and only if R is a right noetherian right V -ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor R -modules is closed under direct summands if and only if R is a right Bass ring. A ring R is said to have no right max-p-middle class if every right R -module is either projective or max-poor. It is shown that if a commutative noetherian ring R has no right max-p-middle class, then R is the ring direct sum of a semisimple ring R 1 and a ring R 2 which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field Q 2 of R 2 has a proper R 2 -submodule which is not complete in its R 2 -topology. Then we show that a commutative noetherian hereditary ring R has no right max-p-middle class if and only if R is a semisimple ring. [ABSTRACT FROM AUTHOR]

In this paper; we will study the relationship between simple submodules of M and the C-ring. A ring R is called C-ring if every torsion module M over R has simple submodules. We investigated that if M is a torsion module over a semisimple ring; so, R is C-ring. Also, we study C-ring when R is Noetherian and T(M) = M such that any direct sum of modules with (SIP) has (SSP) implies that R is C- ring. We proved that if M is a torsion module and M = W + W′, this means R is C-ring where W and W′ are direct sum of M. [ABSTRACT FROM AUTHOR]

Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K0 of R (ignoring the derivation) as an image and contains the reduced K0 of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential projective R modules up to an equivalence. [ABSTRACT FROM AUTHOR]

The core of a finite-dimensional modular representation M of a finite group G is its largest non-projective summand. We prove that the dimensions of the cores of M ⊗ n have algebraic Hilbert series when M is Omega-algebraic, in the sense that the non-projective summands of M ⊗ n fall into finitely many orbits under the action of the syzygy operator Ω. Similarly, we prove that these dimension sequences are eventually linearly recursive when M is what we term Ω + -algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences. [ABSTRACT FROM AUTHOR]

A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules and their allowed morphisms, defined in the paper. The equivalence is expressed by using generators. In particular, we obtain the well-known equivalence of the two categories of projective finitely generated modules; thus, our main result extends this classical one. [ABSTRACT FROM AUTHOR]

In this paper, we characterize rings admitting nonzero homomorphisms between nonzero modules. Also, we characterize rings satisfying the following property: There exists a nonzero homomorphism between any two non-projective modules. [ABSTRACT FROM AUTHOR]

In this investigation we give a module-theoretic counterpart of the well known De Morgan's laws for rings and topological spaces. We observe that the module-theoretic De Morgan's laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan's laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law. [ABSTRACT FROM AUTHOR]

In the paper, criteria are given for the relative projectivity of the -spaces regarded as left Banach modules over the algebra of bounded measurable functions () and the algebra of continuous functions vanishing at infinity (). [ABSTRACT FROM AUTHOR]

The Schröder-Bernstein Theorem for sets is well-known. A dual of the Schröder-Bernstein Theorem is that two sets with surjection maps onto each other are isomorphic. Analogous to this dual, the question of whether two algebraic structures which are epimorphic to each other, are always isomorphic to each other, is of interest. For modules over a given ring, this does not hold true in general. For a ring R , a subclass C of R -modules is said to satisfy the dual of Schröder-Bernstein (or DSB) property if any pair of its members are isomorphic whenever each one is epimorphic image of the other. We investigate the DSB property for the classes of (quasi-)discrete and (quasi-)projective modules among other results in this paper. In particular, we prove that the class of discrete R -modules has the DSB property while the class of quasi-discrete modules does not satisfy the DSB property. On the other hand, over Dedekind domains and generalized uniserial rings, the DSB property holds for the class of quasi-discrete modules. We show that over a right (semi-)perfect ring R , the class of (finitely generated) quasi-projective R -modules satisfies the DSB property, however, over a (von-Neumman) regular ring R these classes do not satisfy the property. We also investigate the DSB property for the class of injective modules. As applications, our investigations provide answers to several open questions posed earlier in this regard. [ABSTRACT FROM AUTHOR]

The history of generalized "stacked bases" theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent: there exists a decomposition G = ⊕ i ∈ I P i into a direct sum of indecomposable modules Pi, such that H = ⊕ i ∈ I (P i ∩ H) ; G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules. [ABSTRACT FROM AUTHOR]

This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring R, we present a characterization of indecomposable projective R-modules and describe a one-to-one correspondence between the projective indecomposable R-modules and the simple R-modules. [ABSTRACT FROM AUTHOR]

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50's. In this paper, we study the notion of a Σ -Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module M is called Σ -Rickart if every direct sum of copies of M is Rickart. It is shown that any direct summand and any direct sum of copies of a Σ -Rickart module are Σ -Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring R is right hereditary if and only if every submodule of any projective right R -module is projective if and only if every factor module of any injective right R -module is injective. Also, we have a characterization of a finitely generated Σ -Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided. [ABSTRACT FROM AUTHOR]

There is a question formulated by R. Ware in 1971. That is, if a projective right R-module P has a unique maximal submodule L, then L is the largest maximal submodule of P. First we give a structure theorem for such a projective module. Next we define the notion of a minimal generator set and give an affirmative answer to this question by using this notion. In the last section, we give an example of a non-quasi-small module to discuss whether the answer to this question is affirmative or negative. The key theorem in Comm. Algebra 31 (2003), pp. 4195-4241 asserts that answer to this question is negative. But our example seems to be a counterexample of this key theorem. [ABSTRACT FROM AUTHOR]

We study the generalized Leibniz brackets on the coordinate algebra of the -dimensional sphere. In the case of the one-dimensional sphere, we show that each of these is a bracket of vector type. Each Jordan bracket on the coordinate algebra of the two-dimensional sphere is a generalized Poisson bracket. We equip the coordinate algebra of a sphere of odd dimension with a Jordan bracket whose Kantor double is a simple Jordan superalgebra. Using such superalgebras, we provide some examples of the simple abelian Jordan superalgebras whose odd part is a finitely generated projective module of rank 1 in an arbitrary number of generators. An analogous result holds for the Cartesian product of the sphere of even dimension and the affine line. In particular, in the case of the 2-dimensional sphere we obtain the exceptional Jordan superalgebra. The superalgebras we constructed give new examples of simple Jordan superalgebras. [ABSTRACT FROM AUTHOR]

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property with respect to\ a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring R and on additional set-theoretic hypotheses. For R commutative noetherian of Krull dimension 0|R|, then the category of all projective modules is κ-accessible. [ABSTRACT FROM AUTHOR]

In Faith [Grundlehren der Mathematischen Wissenschaften. 191 (1976)], Faith asked for what rings R does the Dual Baer Criterion hold in Mod-R, that is, when does R-projectivity imply projectivity for all right Rmodules? Such rings R were called right testing. Sandomierski proved that all right perfect rings are right testing. Puninski et al. [J. Algeb. 484 (2017) pp. 198-206] have recently shown for a number of nonright perfect rings that they are not right testing, and noticed that [Trans. Amer. Math. Soc. 348 (1996) pp. 1521-1554] proved consistency with ZFC of the statement 'each right testing ring is right perfect' (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of nonright perfect rings such that the Dual Baer Criterion holds for all small modules (where small means countably generated, or ≤ 2א0-presented of projective dimension ≤ 1). [ABSTRACT FROM AUTHOR]

Studying the unital simple Jordan superalgebras with associative even part, we describe the unital simple Jordan superalgebras such that every pair of even elements induces the zero derivation and every pair of two odd elements induces the zero derivation of the even part. We show that such a superalgebra is either a superalgebra of nondegenerate bilinear form over a field or a four-dimensional simple Jordan superalgebra. [ABSTRACT FROM AUTHOR]

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G . Let G = G ( q ) be a finite classical group, where q is a p -power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G . We compute, for certain orbit characters and every maximal torus T of G , the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G . Let St denote the Steinberg character of G . Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p -singular elements of G . Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ . Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q -restricted). In addition, for strongly restricted weights μ , we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G . [ABSTRACT FROM AUTHOR]

We study p -Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p -singular elements for a fixed prime p dividing the order of G . Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character φ there exists a minimal p -power, say p a ( φ ) , such that p a ( φ ) φ is quasi-projective. The exponent a ( φ ) only depends on the Cartan matrix of the block to which φ belongs. Moreover p a ( φ ) is bounded by the vertex of the module affording φ , and equality holds in case that G is p -solvable. We give some evidence for the conjecture that a ( φ ) = 0 occurs if and only if φ belongs to a block of defect 0. Finally, we study indecomposable quasi-projective Brauer characters of a block B . This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B . [ABSTRACT FROM AUTHOR]

We show that a commutative monoid A is coexponentiable in CMon(V) if and only if ⊗ A: V → ν has a left adjoint, when V is a cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free. Using a general characterization of the latter, we show that an algebra over a rig or ring R is coexponentiable if and only if it is finitely generated and projective as an R-module. Omitting the finiteness condition, the same result (and proof) is obtained for algebras over a quantale. [ABSTRACT FROM AUTHOR]

In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate our attention on a k--algebra A which is commutative, and use the Kähler differentials module, Ω¹A/k, to define connection (see Subsection 2.1). In this context, it is verified that the existence of connections implies the existence of covariant derivatives (cf. Prop. 2.3), and that every projective module admits a connection (cf. Prop. 2.5). Next (in Section 3), we focus our attention in the discussion of some counterexamples comparing these two notions. In fact, it is known that these two notions are equivalent when we consider regular k-algebras of finite type (see [18], Prop. 4.2). As well as, that there exists a connection on M if, and only if, the Atiyah-Kodaira-Spencer class of M, c(M), is zero (see [17], Prop. 4.3). Finally, we take into account the case where A is (not necessarily commutative) and it is used the bimodule, Ω¹, of noncommutative differentials introduces by Connes ([9], [10]) in place of Kähler differentials to define a connection. In this case, it is proven that a module admits such connection if, and only if, it is a projective module (see [25], Theorem 5.2). [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero, D a locally nilpotent derivation on the polynomial ring k[X1,X2,X3,X4] and A the kernel of D. A question of M. Miyanishi asks whether projective modules over A are necessarily free. Implicit is a subquestion: whether the Grothendieck group K0(A) is trivial. In this paper we shall demonstrate an explicit k[X1]-linear fixed point free locally nilpotent derivation D of k[X1,X2,X3,X4] whose kernel A has an isolated singularity and whose Grothendieck group K0(A) is not finitely generated; in particular, there exists an infinite family of pairwise non-isomorphic projective modules over the kernel A. We shall also show that, although Miyanishi's original question does not have an affirmative answer in general, suitably modified versions of the question do have affirmative answers when D annihilates a variable. For instance, we shall establish that in this case the groups G0(A) and G1(A) are indeed trivial. Further, we shall see that if the above kernel A is a regular ring, then A is actually a polynomial ring over k; in particular, by the Quillen-Suslin theorem, Miyanishi's question has an affirmative answer. Our construction involves rings defined by the relation umv = F(z, t), where F(Z, T) is an irreducible polynomial in k[Z, T]. We shall show that a necessary and sufficient condition for such a ring to be the kernel of a k[X1]- linear locally nilpotent derivation D of a polynomial ring k[X1, . . . , X4] is that F defines a polynomial curve. [ABSTRACT FROM AUTHOR]

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation. [ABSTRACT FROM AUTHOR]

We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups. In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian. The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others. One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian). The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR's homotopy dominating each other have the same homotopy type? [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero, R an affine k-algebra and let Ω(q)(R/k) denote its universal finite Kähler module of differentials over k. In this paper, we consider the tensor, exterior and symmetric algebras of Kähler modules introduced by H. Osborn [9].We explore some interesting properties of the algebras of Kähler modules, which have not been considered before. [ABSTRACT FROM AUTHOR]

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian–Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert C ⁎ -module. An important ingredient in the approach is that twisted group C ⁎ -algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple C ⁎ -algebra has strict comparison of projections. [ABSTRACT FROM AUTHOR]

If n ≡ 0 , 1 m o d 4 , we prove a sum formula V θ 0 (a 0 , a R n) = n ⋅ V θ 0 (a 0 , a R) for the generalized Vaserstein symbol whenever R is a smooth affine algebra over a perfect field k with c h a r (k) ≠ 2 such that − 1 ∈ k × 2. This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension d ≥ 4 over algebraically closed fields of characteristic ≠2. As a consequence, we prove that any projective module of rank 3 with trivial Chern classes over a smooth affine algebra of dimension 4 over an algebraically closed field k with c h a r (k) ≠ 2 , 3 is cancellative. [ABSTRACT FROM AUTHOR]

We analyze the BGG Category O over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a “triangular” GWA for which Category O decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we present a novel connection between blocks of triangular GWAs and Young tableaux, which provides a combinatorial interpretation of morphisms and extensions between objects of the block. [ABSTRACT FROM AUTHOR]

We establish a connection between the integral domains, the projective finitely generated modules over them, the derivations of an integral domain and the (−1, 1)-superalgebras of vector type. We study the properties of the universal associative envelopings of the simple Jordan superalgebras of vector type. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p > 2, and $$\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)$$ be the general linear Lie superalgebra over k. The Cartan invariants for the restricted supermodule category for $$\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)$$ are presented. [ABSTRACT FROM AUTHOR]

In this article we give a bound on the number of generators of a finitely generated projective module of constant rank over a commutative Noetherian ring in terms of the rank of the module and the dimension of the ring. Under certain conditions we provide an improvement to the Forster-Swan bound in case of finitely generated projective modules of rank n over an affine algebra over a finite field or an algebraically closed field. [ABSTRACT FROM AUTHOR]

Discrete (quasi) modules form an important class in module theory, they are studied extensively by many authors. The decomposition theorem for quasidiscrete modules plays an important rule in the better understanding of such modules. In fact, every quasidiscrete module is a direct sum of hollow submodules. Here we introduce some new concepts (weak quasidiscrete, and S- and S-supplemented modules) which generalize the concept of quasidiscrete module. We show that some of the properties of quasidiscrete modules still hold in the class of weak quasidiscrete modules. We also obtain some properties of weak quasidiscrete modules, which are similar to the properties known for quasidiscrete modules. We introduce the concept of generalized relative projectivity (relative S-projectivemodules), and use it to characterize direct sums of hollowmodules. In fact, relative S-projectivity is an essential condition for direct sums of hollow modules to be weak quasidiscrete modules. [ABSTRACT FROM AUTHOR]

We construct some examples of prime Jordan superalgebras of vector type whose odd part is a finitely generated projective module of rank 1 with arbitrarily many generators. These provide some examples of prime Jordan superalgebras of Cheng-Kac type. [ABSTRACT FROM AUTHOR]

Let ( g, [ p]) be a restricted Lie algebra over an algebraically closed field of characteristic p > 0. Then the inverse limits of 'higher' reduced enveloping algebras { u s ( g) | s ∈ ℕ} with χ running over g* make representations of g split into different 'blocks'. In this paper, we study such an infinite-dimensional algebra [Figure not available: see fulltext.] for a given χ ∈ g*. A module category equivalence is built between subcategories of U( g)- mod and [Figure not available: see fulltext.]. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. We prove that, the flat cover of any simple right R-module is projective if and only if R is semilocal and J(R) is cotorsion if and only if R is semilocal and any indecomposable flat right R-module with unique maximal submodule is projective. [ABSTRACT FROM AUTHOR]

We study connection between the Jordan superalgebras of vector type and the finitely generated projective modules of rank 1 over an integral domain. [ABSTRACT FROM AUTHOR]

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M. Homogeneuous ideals of R(M) have the form I(+)N where I is an ideal of R, N a submodule of M such that I M ⊆ N. In particular, [N : M] (+)N is a homogeneous ideal of R(M). The purpose of this paper is to investigate how properties of the ideal [N : M](+)N are related to those of N. We determine when R(M) is a μ-ring, strongly Laskerin ring, Hilbert ring or satisfies Property (U) or Property (FU). It is also shown that if all homogeneous ideals of R(M) have a certain prescribed property, then all ideals of R(M) have the same property. [ABSTRACT FROM AUTHOR]

This note presents some results on projective modules and the Grothendieck groups K 0 and G 0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. [ABSTRACT FROM AUTHOR]

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras. [ABSTRACT FROM AUTHOR]

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described. [ABSTRACT FROM AUTHOR]

Abstract: Let A be a commutative Noetherian ring of dimension d and let P be a projective -module of rank , where . Then [(i)] The natural map is surjective (3.8). [(ii)] Assume is a monic polynomial. Then is an isomorphism (3.8). [(iii)] acts transitively on . In particular, P is cancellative (3.12). [(iv)] If A is an affine algebra over a field, then P has a unimodular element (3.13). In the case of Laurent polynomial ring (i.e. ), (i), (ii) are due to Suslin (1977) , (iii) is due to Lindel (1995) and (iv) is due to Bhatwadekar, Lindel and Rao (1985) . [Copyright &y& Elsevier]

Abstract: We introduce the notion of relative hereditary Artin algebras, as a generalization of algebras with representation dimension at most 3. We prove the following results. (1) The relative hereditariness of an Artin algebra is left–right symmetric and is inherited by endomorphism algebras of projective modules. (2) The finitistic dimensions of a relative hereditary algebra and its opposite algebra are finite. As a consequence, the finitistic projective dimension conjecture, the finitistic injective dimension conjecture, the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for relative hereditary Artin algebras and endomorphism algebras of projective modules over them (in particular, over algebras with representation dimension at most 3). We also show that the torsionless-finiteness of an Artin algebra is inherited by endomorphism algebras of projective modules, and consequently give a partial answer to the question if the representation dimension of the endomorphism algebra of any projective module over an Artin algebra A is bounded by the representation dimension of A. [Copyright &y& Elsevier]

Following [1], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them. [ABSTRACT FROM AUTHOR]

This note can be considered as a supplement to [3] in which the authors characterized noetherian rings by a decomposition property of cyclic modules. We weaken their conditions and obtain the noetherianess for generalized V-rings. [ABSTRACT FROM AUTHOR]

Abstract: We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules. [Copyright &y& Elsevier]

We call a unital locally convex algebra A a continuous inverse algebra if its unit group Ax is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group G on a continuous inverse algebra A by automorphisms and any finitely generated projective right A-module E, we construct a Lie group extension Ĝ of G by the group GLA(E) of automorphisms of the A-module E. This Lie group extension is a "non-commutative" version of the group Aut(핍) of automorphism of a vector bundle over a compact manifold M, which arises for G = Diff(M), A = Cinfin;(M,ℂ) and E = Γ 핍. We also identify the Lie algebra ĝ of Ĝ and explain how it is related to connections of the A-module E. [ABSTRACT FROM AUTHOR]

Let k be a field, H a Hopf k-algebra with bijective antipode, A a left H-module algebra and C a Hopf algebra with bijective antipode which is a left H-comodule algebra. We give sufficient conditions for the category of C-locally finite left A#HC-modules to be semisimple. [ABSTRACT FROM AUTHOR]

In our recent work we investigated ½ (weak) cancellation modules and ½ join principal submodules and showed via the method of idealization most questions concerning these modules can be reduced to the ideal case. The purpose of this article is to continue our study of these modules as well as we introduce and give some properties of the concept of M-join principal ideals. [ABSTRACT FROM AUTHOR]