Showing papers in "Communications in Algebra in 2002"
TL;DR: In this paper, the structures of Armendariz rings and semicommutative rings were studied, which are generalizations of reduced rings, and the classical right quotient rings of Armenderiz rings, the polynomi...
Abstract: In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomi...
242 citations
TL;DR: In this paper, a natural graph associated to the zero-divisors of a commutative ring is considered and the cycle-structure of this graph is classified and some group-theoretic properties of the group of graph-automorphisms are established.
Abstract: There is a natural graph associated to the zero-divisors of a commutative ring In this article we essentially classify the cycle-structure of this graph and establish some group-theoretic properties of the group of graph-automorphisms We also determine the kernel of the canonical homomorphism from to
195 citations
TL;DR: In this paper, a ring is a clean ring if every element of a ring can be written uniquely as the sum of a unit and an idempotent, which is the definition of uniquely clean rings.
Abstract: As defined by Nicholson a (noncommutative) ring is a clean ring if every element of is a sum of a unit and an idempotent. Let be a commutative ring with identity. We define to be a uniquely clean ring if every element of can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ). A ring is a clean ring or uniquely clean ring if and only if is. So every zero-dimensional ring is a clean ring, but a zero-dimensional ring is a uniquely clean ring if and only if is a Boolean ring.
132 citations
TL;DR: In this article, a ring is considered as a module over itself if every fully invariant submodule of the ring is essential in a direct sum-manageable matrix ring.
Abstract: A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.
99 citations
TL;DR: In this article, a strongly regular equivalence is defined and a new characterization of the derived hypergroup of a hypergroup is determined, based on a new strongly regular hypergroup metric.
Abstract: A new strongly regular equivalence is defined and a new characterization of the derived hypergroup of a hypergroup is determined.
96 citations
TL;DR: In this paper, the authors consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented.
Abstract: Let M and N be R-modules. We define where S denotes the class of all M-small modules. We call N an M-cosingular (non-M-cosingular) module if Z M (N) = 0 ( Z M (N) = N). We study the properties of M-cosingular and non-M-cosingular modules in σ[M] We consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented. We also give instances where this torsion theory is cohereditary or splits. Finally we characterise lifting modules N ∊ [M] in terms of Z 2 M (N).
78 citations
TL;DR: In this paper, a simple graph to R with vertices is associated with the set of nonzero zero-divisors of the graph, and two distinct vertices x and y are adjace.
Abstract: Let R be a commutative ring, and let denote its set of zero-divisors. We associate a simple graph to R with vertices , the set of nonzero zero-divisors of . Two distinct vertices x and y are adjace...
75 citations
TL;DR: In this article, it was shown that the result still holds in the more general setting of a polynomial module over a skew polynomial ring, with possibly noncommutative base ring, provided a natural compatibility assumption on (as defined in Sec. 2 below) is fulfilled.
Abstract: In this paper, we extend the recent work of C. Faith[1] in which it is proved, using results of R.C. Shock,[2] that over a commutative ring , the associated primes of the polynomial ring (viewed as a module over itself) are all extended; that is, every may be expressed as , where . This result was originally proved in 1974 by Brewer and Heinzer using localization theory (see[3]), but our work in this paper more closely parallels that of Faith. We will show that the result still holds in the more general setting of a polynomial module over a skew polynomial ring , with possibly noncommutative base ring , provided a natural -compatibility assumption on (as defined in Sec. 2 below) is fulfilled. Moreover, if we assume that , we can make the set of Laurent polynomials into a module over and , and the main result still holds in these cases as well (for -compatible ). Finally, we also show that the result does not extend to the case of skew formal power series rings and skew Laurent series rings in gen...
73 citations
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TL;DR: A commutative ring R with identity is called S-Noetherian, where is a given multiplicative set, if for each ideal I of R, for some and some finitely generated ideal J. as discussed by the authors tie together several different known results.
Abstract: A commutative ring R with identity is called S-Noetherian, where is a given multiplicative set, if for each ideal I of R, for some and some finitely generated ideal J. Using this concept, we tie together several different known results. For instance, the fact that is a Noetherian ring whenever R is so, and that is Noetherian whenever for each nonzero element d of the domain D areboth consequences of the following result: If R is an S-Noetherian ring, then so is , provided for each where S consists of nonzerodivisors.
71 citations
TL;DR: In this paper, it was shown that a level or a half level of the dot-depth hierarchy is decidable if and only if the corresponding level in the Straubing-Therien hierarchy is also decidable.
Abstract: Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions of the operations L ? La and L ? LaA*. Next we turn to concatenation hierarchies. It was shown by Straubing that the n-th level Bn of the dot-depth hierarchy is the variety Vn * LI, where LI is the variety of locally trivial semigroups and Vn is the n-th level of the Straubing-Therien hierarchy. We prove that a similar result holds for the half levels. It follows in particular that a level or a half level of the dot-depth hierarchy is decidable if and only if the corresponding level of the Straubing-Therien hierarchy is decidable.
67 citations
TL;DR: In this paper, the authors define a natural partial order on the orthogonal group and describe the intervals in this partial order, where each subspace of a fixed subspace is represented by a unique orthogonality transformation.
Abstract: We define a natural partial order on the orthogonal group and completely describe the intervals in this partial order. The main technical ingredient is that an orthogonal transformation induces a unique orthogonal transformation on each subspace of the orthogonal complement of its fixed subspace.
TL;DR: In this paper, the dual property of coclosure of a submodule of a module may not always exist and even if it exists, it may not be unique, and the conditions under which a module is a UCC module are investigated.
Abstract: P. F. Smith studied modules in which every submodule has a unique closure and called them UC modules. In this paper we consider modules with the dual property viz., those in which every submodule has a unique coclosure and call such modules UCC modules. Unlike closures, a coclosure of a submodule of a module may not always exist and even if it exists, it may not be unique. We investigate the conditions under which a module is a UCC module. We prove that UCC modules are closed under factor modules and coclosed submodules. We also investigate their properties and their relation to non-cosingular modules, copolyform modules, and codimension modules. We end this paper with the dual of Smith's result on dimension modules.
TL;DR: In this article, the structural properties of stable integral domains of commutative rings with identity have been studied, and the first half of a two-part study has been published.
Abstract: Let be a commutative ring with identity. A regular ideal of is stable if is a projective over its ring of endomorphisms. If every regular ideal of is stable, then is said to be stable. In the first half of a two-part study, we describe the basic structural properties of stable integral domains.
TL;DR: In this paper, an algorithm to construct the automorphism group of a finite p-group is presented, which works down the lower exponent-p central series of the group.
Abstract: We present an algorithm to construct the automorphism group of a finite p-group. The method works down the lower exponent-p central series of the group. The central difficulty in each inductive step is a stabiliser computation; we introduce various approaches designed to simplify this computation.
TL;DR: In this article, the authors consider a commutative Noetherian local ring with maximal ideal and a finitely generated R-module, where the local cohomology modules are known to be Artinian.
Abstract: Let R be a commutative Noetherian local ring with maximal ideal and let M be a finitely generated R-module. The local cohomology modules are known to be Artinian, so is finitely generated for all i...
TL;DR: In this paper, the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra.
Abstract: We offer an approach to basic coalgebras with inspiration in the classical theory of idempotents for finite dimensional algebras. Our theory is based upon the fact that the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra. Our approach is shown to be equivalent to that given by W. Chin and S. Montgomery by using co-endomorphism coalgebras of minimal injective cogenerators.
TL;DR: In this article, the authors considered the problem of computing the Grobner basis of the colon and the radical of a submodule in the ideal case, and provided sufficient conditions to ensure that a sub module has a module-reduced primary decomposition.
Abstract: In this paper, unless otherwise stated, all rings are commutative with identity and all modules are unital. We give sufficient conditions to ensure that a submodule has a module-reduced primary decomposition. In general, the radical of a primary submodule is not prime and the radical does not split intersections of submodules, as is valid in the ideal case. We study sufficient conditions for which these properties hold in the module setting. These conditions involve dimension arguments, consideration of finitely generated modules, and the spectrum of a given prime ideal. Further, we consider the computational problem of finding a Grobner basis of both the colon and the radical of a submodule. A characterization of the elements of the colon is given, along with a method of computing the radical of a submodule in certain cases.
TL;DR: Using toric geometry, lattice theory, and elliptic surface techniques, this paper computed the Picard lattice of certain K3 surfaces and examined the generic member of each of M Reid's list of 95 families of Gorenstein k3 surfaces which occur as hypersurfaces in weighted projective 3-spaces.
Abstract: Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces In particular, we examine the generic member of each of M Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list
TL;DR: In this paper, the lattice of all preradicals on a ring R was studied, and it was shown that it is an atomic and coatomic lattice and described the atoms and coatoms.
Abstract: In this paper we study the lattice of all preradicals on a ring R. We describe this lattice, we prove that it is an atomic and coatomic lattice and we describe the atoms and coatoms. We also give characterizations of simple Artinian, semisimple Artinian, and V-rings in terms of preradicals.
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TL;DR: The notion of left n-coherent rings was introduced in this article, where n-flat and n-absolutely pure modules were used to characterize the left ncoherent ring.
Abstract: The notion of left n-coherent ring (for integers n > 0) is introduced n-coherent rings are characterized in various ways, using n-flat and n-absolutely pure modules
TL;DR: In this article, the second cohomology group of Lie algebras of generalized Witt type is defined, which is more general than those defined by Dokovic and Zhao.
Abstract: We determine the second cohomology groups of Lie algebras of generalized Witt type which are some Lie algebras defined by Passman and Jordan, more general than those defined by Dokovic and Zhao, and slightly more general than those defined by Xu Among all the 2-cocycles, there is a special one we think interesting Using this 2-cocycle, we define the so-called Virasoro-like algebras Then we give a class of their representations
TL;DR: In this article, the authors examined the behavior of strongly FI-extending modules with respect to the preservation of this property in submodules, direct summands, direct sums, and endomorphism rings.
Abstract: A module M is called (strongly) FI-extending if every fully invariant submodule is essential in a (fully invariant) direct summand. The class of strongly FI-extending modules is properly contained in the class of FI-extending modules and includes all nonsingular FI-extending (hence nonsingular extending) modules and all semiprime FI-exten ding rings. In this paper we examine the behavior of the class of strongly FI-extending modules with respect to the preservation of this property in submodules, direct summands, direct sums, and endomorphism rings.
TL;DR: In this article, all endomorphisms of the (finite) Brauer semigroup, its partial analogue and the semigroup of all partitions of a -element set are described.
Abstract: We describe all endomorphisms of the (finite) Brauer semigroup, its partial analogue and the semigroup of all partitions of a -element set.
TL;DR: The Poisson algebras have recently become extremely important in many of the areas f mathematics and have been studied by many people as mentioned in this paper, such as Joseph, Vancliff, Hodges and Levasseur.
Abstract: Poisson algebras have recently become extremely important in many of areas f mathematics and have been studied by many people. In particular, Joseph, Vancliff, Hodges and Levasseur proved that symp...
TL;DR: In this paper, it was shown that the subgroups of the free semigroup on a biordered set in which principal ideals are singletons are free, and an expression was given for the ranks of the maximal subgroups.
Abstract: Easdown has conjectured that the subgroups of the free semigroup on an arbitrary biordered set are free. In this note a weaker conjecture is verified. It is shown that the subgroups of the free semigroup on a biordered set in which principal ideals are singletons are free. In addition, an expression is given for the ranks of the maximal subgroups. This generalizes a result due to Pastijn which involves the free semigroup on a rectangular biset.
TL;DR: In this paper, the graded modules of some Lie superalgebras of Cartan type over a field with character were studied, and they were shown to be sufficiently studied.
Abstract: We know that the graded modules of Lie algebras of Cartan type already are sufficiently studied (see1-4, 6-8 The graded modules of some Lie superalgebras of Cartan type over a field with character
TL;DR: For a selfinjective artin algebra, the projectively stable category of finitely presented left -modules has the structure of a triangulated category and the canonical functor is an equivalence.
Abstract: For a selfinjective artin algebra , the projectively stable category of finitely presented left -modules has the structure of a triangulated category[5] and the canonical functor is an equivalence ...
TL;DR: The Lie algebras of Weyl type were defined and studied in this article, where a commutative associative algebra with an identity element over a field of any characteris...
Abstract: In a recent paper by Zhao and the author, the Lie algebras of Weyl type were defined and studied, where is a commutative associative algebra with an identity element over a field of any characteris...
TL;DR: In this paper it was shown that the Auslander-Reiten quiver has at most four connected components, and is connected if and only if Q has no sink vertices and.
Abstract: Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams , , , , , , , or Q is any of the infinite quivers , , , with , shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] characterising representation-finite hereditary K-algebras (see also [[6], Sec. VIII.5]). It is shown in Sec. 3 that if , then the Auslander-Reiten quiver of the category of finite dimensional left comodules has at most four connected components, and is connected if and only if Q has no sink vertices and .
TL;DR: In this article, it was shown that there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules and prime M-ideals.
Abstract: For a left R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we investigate conditions on the module M which imply that there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules and “prime M-ideals”.