In this talk we consider results and examples of hulls of rings and modules from certain classes including the existence and uniqueness of such hulls. Also we will consider the problem of the termination of an ascending sequence of right essential overrings
These interesting algebras have the property that the Cartan determinant conjecture and its converse hold for them. This talk will compare these algebras with left serial algebras (which also have the property), will show that the Yamagata construction yields quasi-stratified algebras and show other constructions of them.
We discuss consequences of the dual isomorphism theorem and raise some questions.
For a right -module , let be the subcategory of Mod- subgenerated by . A left -module -Mod is - flat, if for any , remains exact. The module is TM-flat, if for any , Tor . A short exact sequence of left -modules is - pure if for any , remains exact.
The theory of flatness and purity relative to the category is developed, including the following. A module is -flat Hom is -injective ( i.e., -injective). For a short exact sequence of left -modules : the connection between the following two properties (a) and (b) is determined. (a) - flat (resp. TM-flat). (b) is - pure.
It was shown that negacyclic codes of length over are precisely the ideals of the chain ring . Using this ring theoritic chain structure, various kinds of distances of all negayclic codes of length over are completely determined. We first calculate the Hamming distances of all such negacyclic codes, which particlularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes over the ring , namely, the -constacyclic codes of length over , where is any unit of with the form . We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes.
Using a variation on the concept of a CS module, we describe exactly when a simple ring is isomorphic to a ring of matrices over a Bézout domain. Our techniques are then applied to characterize simple rings which are either right and left Goldie, right and left semihereditary, or right self-injective.
A classical result due to E.Matlis says that over a left Noetherian ring every injective left module is a direct sum of indecomposable injective left modules. It follows that over the first Weyl algebra every indecomposable injective is the injective envelope of a simple module . Since an arbitrary simple module is or -torsion-free, the ring can be localized at or at to a principal left (and right) ideal domain .
Motivated by a classic treatment of O.Ore, we take advantage of some uniqueness decomposition theorems in and present a nice description of the internal structure of the indecomposable . Furthermore, a consideration of the -action on and of the arithmetic of on an element-by-element basis leads to a new localization of (a subring of the Weyl division algebra). Consequently, we can investigate even further the structure of and determine its socle series. In addition, this extension of allows us to pursue a detailed analysis of the structure of the endomorphism ring and that of the bicommutator of .
[O.Ore, Theory of non-commutative polynomials., "Annals of Math.", 1932, vol.34, 480-508]
It is a well-known result, due to M. Prest and B. Huisgen-Zimmermann and W. Zimmermann, that a ring is of finite representation type if and only if every right -module is of finite length over its endomorphism ring. A ring is called left pure semisimple if every left -module is a direct sum of finitely generated modules. Left and right pure semisimple rings are precisely rings of finite representation type, but it is still unknown whether left pure semisimple rings are always right pure semisimple. In this talk, we discuss a number of characterizations of pure semisimple rings and rings of finite representation type, in terms of endoproperties of their modules, extending the above mentioned result. Our results are closely related with restricted versions of the pure semisimplicity conjecture.
In his monograph/ On Numbers and Games/ [1976], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including -w, w/2, 1/w, ÷w and w-p to name only a few. Indeed, this particular real-closed field, which Conway calls/ No/, is so remarkably inclusive that, subject to the proviso that numbers-construed here as members of ordered "number" fields-be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG, it may be said to contain "All Numbers Great and Small." In this respect,/ No/ bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. However, in addition to its distinguished structure as an ordered field,/ No/ has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or/ simplicity hierarchy/, as we have called it [1994], depends upon/ No/'s (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with/ No/'s structure as an ordered group and an ordered field, respectively, it being understood that/ x/ is/ simpler than y/ just in case/ x/ is a predecessor of/ y/ in the tree. In a number of earlier works [Ehrlich 1987; 1989; 1992], we suggested that whereas the real number system should merely be regarded as constituting an Archimedean arithmetic continuum, the system of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In this paper, we will outline some of the properties of the system of surreal numbers that emerged in [Ehrlich 1988; 1992; 1994; 2001] which lend credence to this thesis, and draw attention to some important respects in which the theory of surreal numbers may be regarded as vast generalization of Cantor's theory of ordinals, a generalization which also provides a setting for Abraham Robinson's [1961] infinitesimal approach to analysis as well as for the profound and largely overlooked non-Cantorian theories of the infinite (and infinitesimal) pioneered by Giuseppe Veronese [1891], Tullio Levi-Civita [1892; 1898], David Hilbert [1899] and Hans Hahn [1907] in connection with their work on non-Archimedean ordered algebraic and geometric systems and by Paul du Bois-Reymond [1870-71; 1882], Otto Stolz [1883], G. H. Hardy [1910; 1912] and Felix Hausdorff [1909] in connection with their work on the rate of growth of real functions.
We develop a method for constructing indecomposable flat cotorsion modules in the category of flat modules over a ring R. This allows to show that the set of these modules is a "cogenerator" of this category. Our methods are inspired by Auslander's work on Representation Theory.
An alternative ring is a ring (not necessarily associative) in which and are identities. A Moufang loop is a loop in which is an identity. An RA (ring alternative) loop is a Moufang loop whose loop ring , for a commutative and associative ring of characteristic different from 2, is alternative but not associative. We show that if is not a Hamiltonian Moufang 2-loop, then , the loop of normalized units of integral loop ring for any RA loop is of central height 1. In case is a Hamiltonian Moufang 2-loop, obviously, is of central height 2.
A ring is said to be right CS if every right ideal is essential in a summand of . A right CS ring is said to be right continuous if any right ideal of which is isomorphic to a summand of is itself a summand of . Right CS rings and right continuous rings are generalizations of right selfinjective rings. It is known that the group algebra of a group over a field is selfinjective if and only if is a finite group. We discuss group algebras that are continuous and give the corresponding conditions on the group . Among others, it is shown that (i) a semilocal group algebra of an infinite nilpotent group over a field of characteristic is CS (equivalently, continuous) if and only if , where is an infinite locally finite -group and is a finite abelian group whose order is not divisible by , (ii) if is a field of characteristic and where is an infinite locally finite -group (not necessarily nilpotent) and is a finite group whose order is not divisible by then is CS if and only if is abelian. Furthermore, a commutative semilocal group algebra is always continuous.
An element of a ring is called clean if it is a sum of a unit and an idempotent. A ring with all elements clean is called clean ring and a module whose endomorphism ring is clean is called a clean module. We will present some recent developements in the theory of clean rings and modules.
in
We define and investigate (structural) row-finite matrices, lower triangular matrices, and power series in near-rings by using inverse limits of some special classes of near-ring matrices. We show polynomials can be embedded in power series and power series can be embedded in lower triangular matrices as in rings. A natural topology is defined on lower triangular matrices by generalizing the concept of order of power series.
A ring is called right annelidan if the right annihilator of any element of the ring is a right waist, i.e. comparable with every other right ideal. Right annelidan rings have a nice characterization within the class of right Bézout rings and within the class of right distributive rings (i.e. rings with a distributive lattice of right ideals). The Bézout or distributive condition enables one to realize a right annelidan ring as a right order in a right uniserial ring. For Bézout or distributive rings with any of a number of mild chain conditions, the annelidan condition is left-right symmetric.
Throughout denotes a commutative ring with identity. is called clean if every element can be written as the sum of a unit and an idempotent. If every proper homomorphic image is clean then is called neat.
We will discuss the class of neat Bézout domains and some of its generalizations. In particular, every such domain is an elementary divisor domain.
A ring is called right Osofsky compatible if a right injective hull of has a ring multiplication which extends the -module scalar multiplication of over . Right nonsingular rings are right Osofsky compatible. A class of right Osofsky compatible rings with will be discussed, where is a maximal right ring of quotients of .
Also we discuss a class of rings such that and there exists a right essential -module extension which satisfies:
(i) there are three ring structures , , and on whose ring multiplications extend the -module scalar multiplication of over .
(ii) and are QF, but not ring isomorphic.
(iii) is not even right FI-extending.
A module is -nonsingular iff is essential in implies , for all . For a module , nonsingular implies polyform implies -nonsingular, while reverse implications are not true. We provide internal characterizations of -nonsingular continuous modules of various types. Our theory properly extends the well-known theory of decomposition of nonsingular injective modules into types, replacing nonsingularity by -nonsingularity, and injectivity by continuity. Our internal characterizations are analogous to the ones obtained by Goodearl and Boyle for the nonsingular injective modules. As a consequence we obtain a characterization of arbitrary -nonsingular continuous modules.
The purpose of this paper is to survey some of the most important results on the theory of weakly injective and weakly projective modules a generalization of injective and projective modules and raise some of the fundamental open problems in this area. It is shown that For a module , there exists a module such that is weakly injective in , for any . Similarly, if is projective and right perfect in , then there exists a module such that is weakly projective in , for any . Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. Among others, For some classes of modules in we study when direct sums of modules from satisfies a property in . In particular, we get characterization of locally countably thick modules a generalization of locally modules. Characterizations of rings over which every weakly injective is weakly projective and conversely are given. Finally, we conclude with several open questions. The following are some of these problems:
(joint work with Hans-Dietrich Gronau, Rostock University, Germany)
Let be the complete oriented graph on the finite set . A family of spanning subgraphs is said to be an orthogonal cover if (1) every arrow in occurs in exactly one of the and (2) for any pair , the graph and the opposite of the graph have exactly one arrow in common.
Orthogonal covers are frequently studied in combinatorics. Particular interest is in covers which are polycyclic (each is a disjoint union of cycles) and isophyllic (all the are isomorphic as graphs).
It turns out that endomorphisms of finite modules yield a rich variety of orthogonal covers; suitable choices for the endomorphisms produce covers which are both polycyclic and isophyllic.
A commutative local ring of finite embedding dimension has a Noetherian completion. This allows us to apply some tools from commutative algebra to this larger class. In particular, we get primary decomposition for closed ideals. If the ring is moreover geometric, that is to say, has the same (Krull) dimension as its completion, then the lattice of ideals is well-behaved. This leads to several Noetherianity criteria for such rings.
Diagram algebras are associative algebras having a basis consisting of diagrams. Partition algebras arose in the context of Potts model in statistical mechanics and as generalization of Temperely Lieb algebras. The diagram algebras include partition algebras, Brauer algebras, Temperely Lieb algebras, planar algebras and the group algebras of the symmetric group.
The study of diagram algebras has proved to be valuble to both ring theorists and physicsts(cf.Paul Martin, Temperely Lieb algebras for non-planar statistical mechanics- the partition algebra construction, J.Knot Theory and Ramifications., Vol 3,No 1 ( 1994), 51-82). These diagram algebras have shown to have strong connection with Lie theory.
A new class of diagram algebras namely Signed Brauer Algebras, G-Brauer algebras and Vertex colored Partition algebra havae been introduced by us. The structure and representation of the above algebras will be discussed and the Schur-Weyl duality will be explained in detail
It is well known that a ring in which each right ideal is a finite direct sum of injective right ideals is semisimple artinian. We introduce rings in which each right ideal is a finite direct sum of quasi-injective right ideals. Such rings will be called right finitely - rings. These rings are a natural generalization of right -rings, which are defined as rings in which each right ideal is quasi-injective. We characterize various classes of right finitely - rings and give some examples.
Let be a Galois algebra over a commutative ring with Galois group such that the subalgebra of the elements fixed under the elements of a subgroup is separable. If satisfies the fundamental theorem for Galois extensions, then is one of the following classes: (1) indecomposable commutative, (2) a direct sum of and for some minimal central idempotents and , and (3) indecomposable such that the commutator subalgebra of each separable subalgebra is a direct sum of certain projective submodules of induced by .
Let be a coalgebra over a field . If is a left quasi-coFrobenius coalgebra, then is generator for the category of left comodules. We show that the converse is true if has a finite coradical series.
Linear codes defined over a finite Frobenius ring have an extension property--any linear isomorphism between codes that preserves Hamming weight necessarily extends to a monomial equivalence. The converse is also true--if linear codes over a finite ring have the extension property, then the ring is necessarily Frobenius. This talk will describe the main ideas in the proof of the converse, following a strategy of Dinh and Lopez-Permouth.
A ring R is called Baer (resp. quasi-Baer) if the left annihilator of any nonempty subset (resp. any ideal) of R is generated by an idempotent. It is unclear when a group ring is (quasi-) Baer. In this talk, we present recent progress towards this question.