In this joint work with Aaron Gulliver, we investigate certain construction methods for selfdual ternary codes using a class of conference matrices.Examples are given for lengths up to 96. The codes of length 12, 24, 36 and 48 are equivalent to the Pless symmetry codes. In addition, we have determined that the minimum distance for the codes of length 60 and 84 are the same for both classes. The minimum distance calculations for length 96 (dimension 48) yields 24, improving the previously known bounds according to Brouwer's tables. For the case of length 84, we have determined that the minimum distance of the Pless symmetry code is exactly 21. The relationship between our codes and those of Pless (symmetry codes) is also examined.
Recent results have revealed that Dowling geometries share a number of the nice properties of projective geometries; in some sense the former may be regarded as group-theoretic analogues of the latter. A fundamental result of Rado concerning projective geometries states that every finite geometry representable over a field is representable over a finite field. The corresponding question (posed by Bonin) for Dowling geometries is the following:
Is it true that a finite geometry that embeds in the Dowling geometry of an infinite group, necessarily embeds in a Dowling geometry of a finite group?
In this talk we demonstrate, constructively, that the above question has a negative answer. By way of commentary on the analogy between Dowling and projective geometries, we observe that although our findings contrast with Rado 's result for projective geometries, it is a purely group-theoretic phenomenon tha t settles the matter. This is joint work with Hongxun Qin, Edmund Robertson, and Ákos Seress.
Let be an -dimensional vector space over , where , is a prime. For , let be the (0,1)-incidence matrix with rows and columns respectively indexed by the - and -dimensional subspaces of , and with -entry equal to one if and only if the -dimensional subspace is contained in the -dimensional subspace . The rank of was computed by Yakir and Frumkin over any field not of characteristic . In this talk, we extend their results to determine the -adic Smith normal form of for .
Applying results from quadratic forms, projective geometries, and recent results of Brouwer and van Dam, we construct the first known amorphic association scheme with negative Latin square type classes where the underlying group structure is not elementary abelian. We give a simple proof of a result of Hamilton that generalized Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.
This represents joint work with Qing Xiang.
We will describe some new constructions of -difference matrices, the cyclic group of order , divisible by neither 2 nor 3.
A graph is path-perfect if there is a positive integer such that the edge set of the graph can be partitioned into paths of length We prove Fink and Strait's conjecture: A complete bipartite graph on vertices is path-perfect if and only if there is a positive integer such that the following two conditions are satisfied; (i) and (ii) This is a joint result with W. Cao.
Two linear codes in are called equivalent if one can be obtained from the other through the actions of a monomial matrix and an automorphism of . Let be the number of nonequivalent -dimensional codes in . We describe a method for computing . The method relies on the canonical forms of semi-linear transformations of , which are characterized by indecomposable skew polynomials over . Some numerical results will be presented.
Aiso Heinze, in his Ph.D Thesis (Oldenburg, Germany, 2001), has determined all partial difference sets (briefly pds's) over groups of order up to 49. In his research he was using the computer catalogues of strongly regular graphs with a small number of vertices (Ted Spence, University of Glasgow, Scotland),
Among his results was the proof of the existence of two unusual pds's over groups of order 36, one of these groups is a direct product of two copies of dihedral groups of order 6. Both these pds's imply the same (up to isomorphism) strongly regular graph with the parameters , , . The automorphism group of has order 648, it acts transitively on the vertex set of . Moreover, it turns out that is a Latin square graph, coming from a proper loop of order 6. (A loop is called proper if it does not contain a group in its main class.) Soon after we realized that A.Barlotti and K.Strambach were looking for exactly such example of a proper loop (Advances in Mathematics, 49:1-105, 1983).
We managed to get a quite beautiful computer free interpretation of our result. Moreover, starting from our construction, we were able to prove an existence of an infinite series of proper loops with similar properties. Our loops have order , where is a prime, equal to 3 modulo 4. Finally, we will discuss links between our results and various investigations of many other mathematicians, including: E. Schoenhardt, A. Sade, R.A. Bailey, C.E. Praeger, A. Sprague, E. Wilson, R.L. Wilson Jr., E.G. Goodaire & D.A. Robinson, K. Kunen.
In our research we were using computer packages COCO, GAP, GRAPE, nauty.
Using Tree Decompositions and Robertson-Seymour's Excluded Minor Structure Theorem, we show that a linear (in terms of ) bound on the connectivity implies that sufficiently large graphs will contain and minors. In particular, for any , any -connected sufficiently large graph must contain a -minor. Moreover, for any , any -connected sufficiently large graph myst contain a -minor. The conjectured connectivity is .
This is joint work with Thomas Boehme, Ken-ichi Kawarabayashi and Bojan Mohar.
Hadwiger's Conjecture states that the chromatic number of any graph is at most the size of the largest clique minor the graph possesses. It is believed that if a counterexample for the conjecture can be found, it should appear in the case of graphs with stability number equal to two, since they have large chromatic number. On the other hand, if the conjecture holds for this case, we obtain further proof that it should hold in general. In this talk, we establish the Hadwiger Conjecture for graphs with stability number two which do not contain induced cycles of size 4 or 5.
Utilizing classical elliptic function invariants, we first sketch our derivation of several useful new formulas for Ramanujan's function. This work includes: the main pair of new formulas for the function that ``separate'' the two terms in the classical formula for the modular discriminant, a generating function form for both of these formulas, a Leech lattice form of one of these formulas, and a triangular numbers form. We then present analogous new formulas for several other classical cusp forms that appear in quadratic forms, sphere-packings, lattices and groups.
A binary code with the same weight distribution as its dual is called formally self-dual(f.s.d.). We consider only even f.s.d. codes (all weights even). We show that if C is a Type l (not Type ll) s.d.[n,n/2,2[n/8]]code,with 8/n, then its weight distribution is unique. We determine the possible weight enumerators of a near-extremal f.s.d. even [n,n/2,2[n/8]] code with 8/n as well as the dimension of its radical.
This is joint work with Jon-Lark Kim.
We prove that the only geometry of rank all of whose proper contractions are ternary Dowling geometries is the ternary Dowling geometry. We use this to prove a stronger version of a conjecture of Joseph Kung and James Oxley.
This is joint work with Tom Dowling.
Vortex structure is a fundamental feature of graph minor structure theory. Perhaps it is also the least understood by the majority of graph theorists. This lecture will be an explanation of what is a vortex and how vortices fit into the structure theory and well-quasi-ordering aspects of graph minor theory.
Back in 1865, Sylvester posed the following fundamental (and still open) problem in Geometric Probability. For a probability distribution in the plane, let denote the probability that four independent -random points form a convex quadrilateral. What is Sylvester Four Point Problem's Constant ? (where the infimum is taken over all probability distributions with positive Borel measure). It is known that the problem of finding is closely related to the problem of determining the minimum number of convex quadrilaterals in a finite set of points in the plane in general position. This problem is in turn closely related to the rectilinear crossing number of the complete graphs , and also to the problem of determining the number of -sets in a finite set of points in the plane in general position. Recently, we have refined the best lower bound for , and have shown that . This is joint work with József Balogh.
The function lattice, or generalized Boolean algebra, is the set of -tuples with the th coordinate an integer between 0 and a bound . Two -tuples -intersect if they have at least common nonzero coordinates. We prove a Hilton-Milner type theorem for systems of -intersecting -tuples.
This is joint work with P. L. Erdos and L. Székely.
A signed graph is a pair in which is a graph and is a labeling of the edges with elements of the multiplicative group . A circle in a signed graph is called negative if the product of labels on its edges is negative. Otherwise the circle is called positive. A signed graph is called balanced if all its circles are positive.
There is a matroid associated with whose elements are the edges of . The rank of a subset of edges is equal to the number of vertices incident to edges in minus the number of balanced components in the subgraph defined by the .
In this talk we will discuss several structural properties of this signed-graphic matroid: representability, topology, connectivity, and decompositions.
This talk presents joint work with Hongxun Qin of The George Washington university.
The problem of generating frequent itemsets arises in the field of Data Mining. In this problem we are given a data file that may be regarded as a collection of subsets of some finite universal set. Each member of the universal set is called an "item". A "frequent itemset" is a set of items that appears at least times in the data file (for some threshold ). An important problem in Data Mining is to compute all frequent itemsets, and numerous algorithms for this have been given.
These algorithms are usually evaluated empirically - their relative execution time on various data files is reported. Analysis by computational complexity does not reach much beyond the observation that the worst case execution time is exponential (at least), since output size is exponential in terms of input size. We describe a variant of one well known algorithm to compute frequent itemsets, and sketch a proof that, when , its execution time is linear in terms of the sum of output size and input size.
We consider the simple group and its automorphism group each in its classical action on 9 points. As these actions are known to be non-geometrical in the sense of D.Betten, we introduce a new notion, ``geometrical group of second order," which allows us to interpret our groups and of degree 9 (as well as their corresponding stabilzers of degree 8) as automorphism groups of suitable collections of regular uniform incidence structures.
Along the way we are led to consider so-called overlarge sets of Steiner designs, in particular the overlarge set of nine Steiner 3-designs with parameters which was initially found by W.L. Edge and later rediscovered by D.R. Breach and A.P. Street. From here, we give a simple new construction of a classical partial geometry with parameters having 120 points, 135 lines and automorphism group . We also provide new constructions of the two partial geometries with parameters which were characterized by R. Mathon with the aid of a computer.
This is joint work with M. Klin and S. Reichard.
Smith normal form is a generalization of -rank. We use -adic Jacobi sums, a theorem of Daqing Wan, and some representation of the general linear group to determine the Smith normal forms of designs arising in finite Desarguesian geometries.
Let be an odd prime. Let be a finite -group and be an irreducible character of . Denote by the complex conjugate of . Assume that . We show that the number of distinct irreducible constituents of the product is at least .
Subgroups and of a finite group are said to be totally permutable if every subgroup of permutes with every subgroup of . The behaviour of finite pairwise totally permutable products are studied with respect to certain classes of groups including the class of groups where all the subnormal subgroups permute with all the maximal subgroups,the so called -groups and also the class of all groups where all the subnormal subgroups are permutable,the so called -groups (joint work with Peter Hauck and Hermann Heineken).
For a fixed positive integer , define the Neumann -center of the group to be the set of all elements of such that for all -subsets of . Define the Freiman center to be the set of all in such that for each in , the set has at most three distinct elements. We discuss two questions:
Dowling geometries are a class of finite geometries defined using groups, which share many of the nice properties of projective geometries. A fundamental result of Rado concerning the latter type of geometries states that every finite geometry representable over a field is representable over a finite field. The corresponding question for Dowling geometries is the following:
Is it true that a finite geometry that embeds in the Dowling geometry for an infinite group, necessarily embeds in a Dowling geometry for a finite group?
This question can be reduced to a purely group-theoretic one, which may be stated roughly as follows:
Is it true that every finite partial Cayley table of an infinite group arises as a partial Cayley table of a finite group?
It is demonstrated that this latter question has a negative answer, thereby settling the original embeddability question. This is joint work with Honqxun Qin, Edmund Robertson, and Akos Seress.
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Given an -dimensional vector space over the field of complex numbers and a nondegenerate hermitian form, we consider the set of subspaces of that are either positive-definite or have a further positive-definite subspace of maximal dimension. We define a rank-n geometry on this set (using symmetrized containment as our incidence relation), demonstrate the simple-connectedness of this geometry in most cases, and pose some questions about the automorphism group of this geometry.
For a group, a mapping is called a complete mapping of if the mappings and are both bijections.
We derive a sufficient condition for a subgroup of , the algebraic closure of , to admit complete mappings. We will use this condition to give new proofs of the existence of complete mappings for some classes of linear groups.
We say a subgroup property is closed under joins if and implies that . Then if is closed under joins, given any subgroup of a group , the core of in , , can be defined to be and , the unique maximal subgroup of that has property in . Note that if means is normal in , then , the normal. We say a subgroup property is closed under intersections if and implies that . Then if is closed under intersections, given any subgroup of , the closure of in , , can be defined to be the intersection of all subgroups of such that and ; this is the unique minimal subgroup of that contains and has property in . Note that if means is normal in , then , the normal. We say a subgroup property is a lattice property if it is closed under joins and closed under intersections, so that i Theorem 1. Suppose is closed under joins, is closed under intersections, and is a lattice property. Suppose further that if , then and . Then the following are equivalent: When (a) or (b) holds, so that both hold, we will say and are -supplementary. We look at a variety of aspects of this concept, including, given a join-closed , the existence and determination of the unique weakest intersection-closed property, denoted , such that and are -supplementary. We say and , are -complementary. Similarly, given an intersection-closed , we determine the unique weakest join-closed property, , such that and are Theorem 2. Suppose is in the image of and is in the image of . Then if and only if .
In this talk we investigate the following problem: properties which transfer (or do not transfer) from all cyclic subgroups, or all abelian subgroups to all arbitrary subgroups. We solve this problem completely when is the property of having finite index in its normal closure, proving that carries from abelian, but not from cyclic to arbitrary subgroups. We use primarily some results of B.H. Neumann.
This talk does not settle the issue of the existence of such groups. Assuming a first-order language, , appropriate for group theory, the universal theory of a class of groups is just the set of all universal sentences of true for every group in the class. Sometime ago Verena Huber Dyson asked whether or not the universal theory of torsion groups coincides with the universal theory of finite groups. Here we show that if these theories coincide, then there cannot exist a finitely presented group of prime exponent. We cannot, however, say anything about the converse.
A subgroup of a group is called 2-subnormal, or subnormal of defect 2, if there exists a normal subgroup such that is normal in . Heineken, Stadelmann, and Mahdavi have extensively studied groups having certain collections of subgroups subnormal of defect 2. Mahdavi looked at the interdependencies between groups having all cyclic, all abelian, all class 2, and all subgroups 2 subnormal. He left as an open question whether an arbitrary group having all class 2 subgroups 2-subnormal implies that all subgroups are 2-subnormal. We construct a group having all class 2 subgroups 2-subnormal, but not having all subgroups 2-subnormal.
We establish a correlation between the structures of a group of power automorphisms of some group and their mutual commutator subgroup and consider the consequences for the norm of a group, and for its capability. (Report on work done by J.Beidleman, H.Heineken and M.Newell)
in
This is joint work (in progress) with R Guralnick. We will prove that if is a nonparabolic element in a classical simple group then 3 conjugates of generate . The methods used will be probabilistic in nature relying on Aschbacher characterization of maximal subgroups of classical groups.
The study of 2-characters of a group by Sehgal and the presenter led to the introduction of weak Cayley table maps (WCT maps) between groups.
A map is a WCT-homomorphism if
The group of WCT isomorphisms from a group to itself has been studied by Humphries, who proved that for symmetric groups and free groups it is generated by the automorphisms and the map . A WCT-isomorphism between two groups ensures that they have the same character table and other properties.
We call a map is a conjugacy-fuzzy (cf)-homomorphism if
The weaker property that defines fc maps has produced some surprising results. The cf-isomorphisms need not preserve conjugacy classes and can exist between groups with different character tables. Any permutation which fixes the conjugacy classes of a group setwise is a cf-isomorphism. If we consider the cf-isomorphisms from to itself there are three kinds of elements.
(1) Permutations which fix each conjugacy class.We call such permutations basic standard cf isomorphisms.
(2) cf-isomorphisms which permute conjugacy classes but do not split them.
We call the subgroup of permutations generated by cf-isomorphisms of types (1) and (2) the standard group.
(3) There are non-standard cf isos.
The set of all cf-isomorphisms is not necessarily a group but is a union of cosets of the standard group. We can characterize cf-isomorphisms by the property
We do not, however, have a more intrinsic group theoretical or character theoretical characterization.
Many questions can be posed on the lines of: which groups have only standard cf maps, do free groups have non-standard maps, what equivalence relation does ``having a cf-isomorphism between them'' produce on the set of -groups.
It is well known that the squares of elements in a group do not form a subgroup and that the alternating group on four letters is minimal with this property. For given , what is the group of minimal order such that the powers of elements do not form a subroup? For odd , it can be shown that the dihedral group of order is a minimal counter-examle, where is the smallest prime dividing .
If is even, the situation is more complex. The order of the minimal counter example depends on the odd prime factors of and the exact -power dividing .
Let be a Hall system of a finite, solvable group and let denote the set of all -permutable subgroups of . It is known that is a sublattice of the subgroup lattice of . We call an -group if the lattice is modular. (The conjugacy of any two Hall systems implies that the condition is independent of the Hall system chosen).
It is evident that if for all , , then is modular. Our principal result is that the converse holds: If , then any two -permutable subgroups permute with each other. In addition to the proof of this result, we will discuss closure properties of the class and local-global results (about what happens at each prime).
Let be a finite group, and let be the character degrees of . We consider the graph whose vertex set is the set of primes that divide degrees in . There is an edge between and if divides some degree in . These graphs have been studied in a number of places. In this paper, we classify the graphs that can arise when is a solvable group of Fitting height 2.
When is a finite nonabelian group, we associate the common-divisor graph with by letting nontrivial degrees in be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set of vertices for this graph is said to be strongly connective for if there is some prime which divides every member of , and every vertex outside of is adjacent to some member of . When has a nonabelian solvable quotient, we show that if is connected and has a diameter of at most , then indeed has a strongly connective subset.
For a given prime , what is the smallest integer such that there exists a group of order in which the commutators are not a subgroup? In this paper we show that for any odd prime and for (Joint work with Luise-Charlotte Kappe).
Weisner's group theoretic method is applied to obtain new generating functions for Hermite matrix functions defined by L.Jodar and R.Com-pany, by giving suitable interpretation to the index . The algebraic structure underlying Hermite matrix functions(HMF) can be recognized in full analogy with Hermite functions(HF), thus providing a unifying view to the theory of both HMF's and HF's. A few special cases of interest are also discussed.
A subgroup of a finite group is said to have the cover-avoidance property in and is called a CAP-subgroup of if either covers or avoids each chief factor of . It is not difficult to show that the cover-avoidance property is neither persistent nor transitive, but various properties of CAP-subgroups can be obtained by considering weaker forms of persistence and transitivity. In particular, for a CAP-subgroup of , we say that is CAP-transitive in if each CAP-subgroup of is a CAP-subgroup of , and we say that is CAP-persistent in if each CAP-subgroup of contained in is a CAP-subgroup of .
Let -groups, -groups, and -groups be those groups in which, respectively, normality, permutability, and Sylow-permu-tability are transitive. We say is a -group if is a -group, while is a Hall- group if contains a normal nilpotent subgroup such that is an -group. Some basic results on -groups, Hall- groups, Hall- groups, and Hall- groups will be discussed along with some characterization theorems (in the finite solvable case). A key step in characterizing the above groups is showing that they have a nilpotent hypercommutator which is also a Hall subgroup. Also key is the result that being a -group implies is in fact a -group.
Preliminary report.
Generalizing a result for finite groups from Graham Ellis and Frank Leonard (1995) to polycylcic groups, we are able to apply both hand and computer methods to compute the nonabelian tensor square of the free nilpotent group of class 3 and rank . Using this approach we will also reproduce results from our paper on the nonabelian tensor square of 2-Engel groups of rank to appear in Proceedings of the Edinburgh Mathematical Society (joint work with Russell D. Blyth and Robert F. Morse).
Fix any prime and positive integer . Let denote the cyclic group of order . For each integer , define to be the regular wreath product group . We mention that is isomorphic to a Sylow -subgroup of the symmetric group , and that for each prime power such that is the full -part of , the group is isomorphic to a Sylow -subgroup of if and only if .
We determine the ascending central series (and in the process, the nilpotence class) of , which has a very nice description. For , its nilpotence class is . Using similar methods, we also determine the ascending central series and the nilpotence class of the regular wreath product group , where is elementary abelian of order , for an arbitrary positive integer . We determine that this last group has nilpotence class .
We will define a geometry on which an orthogonal group acts. We will show that this geometry is simply connected and using Tits' lemma we will be able to identify with the universal completion of an interesting amalgam of its subgroups.
Outer automorphism groups of fundamental groups of surfaces of genus are known to be residually finite. These groups can be considered either as generalized free products or 1-relator groups. It is natural to ask which generalized free products or 1-relator groups have residually finite outer automorphism groups. We prove certain generalized free products have Property , from which we derived that certain tree products of groups have residually finite outer automorphism groups. We apply these results to groups of linkages, hose knots and 1-relator groups with non-trivial centers (joint work with G. Kim).
It is known that in a finite solvable group , a subgroup is abnormal if and only if every subgroup of containing is self-normalizing in . Although, in general, the assumption of solvability cannot be dropped, in this talk we discuss a proof of the theorem for the special case and a second maximal intransitive subgroup of . Qinhai Zhang has previously spoken on this topic at these meetings in the case where is a power of a prime (joint work with Ben Brewster, Binghamton University and Qinhai Zhang, Shanxi Teachers University).
Professor Kappe has studied the power margin in groups. One goal of this research is to find conditions on groups which force the powers of elements to behave in an aesthetically pleasing way. For -groups, these three properties of regular -groups are generally considered desirable:
Antonio Behn proved that if is prime ring with polycyclic-by-finite, then is torsion-free or and . In this paper we prove that if is semiprime ring having no domains summands with polycyclic-by-finite, then is hereditary. Description of such group algebra, when is algebraically closed, is also given (JoinT work with S. K. Jain and J. B. Srivastava).
A right -module is said to be subdirectly irreducible if it is nonzero and has a smallest nonzero submodule, or equivalently, if has a simple essential socle. Since this concept is dual to that of cyclic module, subdirectly irreducible modules are also called cocyclic. A submodule of is said to be a subdirectly irreducible (or cococyclic) submodule of if the quotient module is subdirectly irreducible. These concepts can be naturally relativized with respect to a hereditary torsion theory on Mod-, or more generally, can be defined for arbitrary posets with 0 and 1.
As a module is the sum of all its cyclic modules, dually, for any module , the intersection of all subdirectly submodules of is zero; one also says that the lattice of all submodules of is rich in subdirectly irreducibles. The relative version of this property does not hold in general.
In this talk we characterize right modules having relative -dual Krull dimension less than or equal to a given ordinal. This requires that is rich in -subdirectly irreducibles. Therefore, we first look for sufficient conditions on the torsion theory to insure that, for any module , the lattice of all -closed submodules of is rich in subdirectly irreducibles. In particular we give a relative version to an extension in the more general setting of dual Krull dimension of a nice recent result due to Carl Faith saying that a right -module is Noetherian if and only if is QFD and satisfies the ACC on subdirectly irreducible submodules.
The results to be presented, have been obtained jointly with Mihai Iosif (Bucharest University, Romania) and Mark L. Teply (University of Wisconsin-Milwaukee, USA).
A ring is called right (left) max-min -ring if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of is a direct summand of . In this paper we show that if is a prime ring which is not a domain, then is right nonsingular, right max-min -ring with uniform right ideal if and only if is left nonsingular, left max-min -ring with uniform left ideal. The above result gives, in particular, Huynh-Jain-López Theorem [Proc. Amer. Math. Soc. 128, 3153-3157 (2000)] (joint work with Adel N. Alahmadi and S. K. Jain).
(Preliminary report)
We investigate modules which satisfy an analog in of the "condition H" introduced in Gabriel's thesis.
Fifteen years ago, Bell and Guerriero conjectured that a ring having only a finite number of noncentral subrings must be either finite or commutative. We have recently found a proof of this result, which involves some unexpected notions. We give an indication of the proof, and we discuss some related results.
(This talk is based on joint work with Jae Keol Park and S. Tariq Rizvi).
In this talk we discuss various concepts for the hull of a ring or module, where the hull is from a certain class of rings or modules, respectively. Existence and/or uniqueness results on hulls from various classes (including extending, FI-extending, continuous, Baer, etc.) are presented. Also the transfer of information between an object and its hull is considered. Examples are provided to illustrate the theory.
We consider conditions that make the rings in the title Noetherian.
A type decomposition of a module over a ring is a direct sum decomposition for which any two distinct summands have no nonzero isomorphic submodules. In this paper, we investigate when a module possesses certain kinds of type decompositions and when such decompositions are unique.
A class of right -modules of modules is a type (or natural class) if it is closed under isomorphic copies, submodules, arbitrary direct sums and injective hulls. A submodule of is a type submodule if, for some type , is a submodule of which is maximal with respect to . In this case, we also say is a type submodule of type . Just as every submodule has a complement closure in , so also it has a type closure . Just as in the complement case, need not be unique. A module satisfies UTC (unique type closure) if every submodule has a unique type closure , or equivalently if for every type , has a unique type submodule of type . Several equivalent characterizations of UTC modules are obtained. Fully invariant type submodules and fully invariant type direct summands are explored.
Galois rings have been used widely in Algebraic Coding Theory. In this talk, we will discuss negacyclic codes of length over Galois rings. It will be shown that is indeed a chain ring, and hence obtaining the complete structure of negacyclic codes of length over , as well as that of their duals. We also address the minimum Hamming weights of such negacyclic codes. Open directions for further investigations will also be discussed.
A right module over a ring (with identity) is called endofinite if it has finite length as a left module over its endomorphism ring. It is well-known that endofinite modules have many attractive properties: Every endofinite module is -pure-injective, with nilpotent Jacobson radical of the endomorphism ring End, and admits an indecomposable decomposition , where the number of isomorphism classes of the is finite. In this talk, we will discuss modules satisfying some weaker forms of chain conditions on endosubmodules, and show that a number of the above properties of endofinite modules can be extended to our situation. Among the applications, we show that any pure-injective pure-projective module over an Artin algebra is endo-Artinian. (A part of the results of this talk is based on our joint work with Sergio R. López-Permouth).
It is proved that a ring is right -pure injective in the category of flat modules if and only if it is right perfect. Several other conditions are shown to be equivalent. For example, that a countable direct sum of copies of its pure-injective envelope in Flat- is pure-injective in that category. (Joint work with Ivo Herzog).
Many arguments in the Theory of Rings and Module are, on close inspection, purely Lattice theoretic arguments. Grigore Calugareanu has a long repertoire of such results in his book "Lattice Concepts on Module Theory". The Levitski-Hopkins Theorem is interesting from this point of view, because only part of it lends to an obvious lattice theory approach. The rest of it requires a new construction, the Hyper-radical, which will be presented in this talk.
It is shown that a ring is right cotorsion if and only if it is the endomorphism ring of a pure-injective object in a locally finitely presented additive category. It is also shown that these rings are (von Neumann) regular and self injective module their Jacobson radical and idempotents lift. (Joint work with Pedro A. Guil Asensio).
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. The group algebra is principally injective if and only if is a locally finite group. We discuss group algebras that are continuous (or CS) and give the corresponding conditions on the group . Among others, it is shown that if the group algebra of a group over a field is right continuous then is a locally finite group. In particular, a right continuous group algebra is principally injective.
We investigate in this paper von Neumann regularity of certain rings whose simple singular left -modules are -injective. As a nontrivial generalization of a reflexive ring, idempotent reflexive ring is considered. It is proved that if is an idempotent reflexive ring whose simple singular left -modules are -injective, then for any nonzero element in , there exists a positive integer such that and . Several known results are extended and unified.
In joint work with L. Levy, we showed that a commutative, Noetherian ring has tame representation type if and only if is a Klein ring or a homomorphic image of a Dedekind-like ring.
Klein rings have finite length, while Dedekind-like rings are reduced and one-dimensional and have the property that all of their finitely generated indecomposable modules have torsion-free rank at most two. Following this observation, R. Wiegand conjectured that, for a commutative, Noetherian, reduced, one-dimensional ring , there is a bound on the torsion-free rank of finitely generated indecomposable -modules if and only if has tame representation type (if and only if is a Dedekind-like ring).
In joint work with W. Hassler, R. Karr, and R. Wiegand, we prove this conjecture and more.
We say a ring is "right annelidan" if the right annihilator of any element of the ring is a right waist. This class of rings encompasses domains as well as right uniserial rings and certain other local rings. We obtain results on the structure of right annelidan rings, their ideals, and important radicals. We show the Köthe Conjecture holds for this class of rings, and we study the relationship between this and various other major classes of rings (joint work with Ryszard Mazurek).
We discuss a right injective hull of a ring such that has exactly four distinct ring structures whose multiplications extend the module multiplication of over and all these ring structures are isomorphic. This gives a negative answer to Osofsky’s question on the uniqueness of the compatible ring multiplication for the injective hull. Various types of ring hulls of and all intermediate ring structures between and are discussed. (jointly done with Gary F. Birkenmeier and S. Tariq Rizvi).
We discuss old and prove new results on rings over which every projective right module is a direct sum of finitely generated modules.
By Albrecht and Bass this is true for left or right semihereditary rings. It follows from a generalization of Krull-Azumaya-Schmidt theorem that semiperfect rings enjoy this property. By Harada the same is true for every commutative weakly noetherian ring.
We give a precise criterion (in terms of matrices over a ring) when every projective right module is a direct sum of finitely generated modules.
Outside the above mentioned classes of rings, the answer to this question is widely unknown. For instance, it is not known if every projective module over a commutative domain is a direct sum of finitely generated modules.
In particular, we give different examples of noetherian rings where the answer to the above question is 1) negative; 2) positive; and 3) unknown.
For instance, as a byproduct of our results we give a complete classification of non-finitely generated projective modules over all primitive factors of ( is an algebraically closed field of characteristic zero) and over the ring of differential operators on , the projective space.
The following statement is known as Vaught's conjecture:
Conjecture. Let be a countable complete first-order theory. If has uncountably many non-isomorphic countable models, then has non-isomorphic countable models.
Garavaglia proved Vaught's conjecture for -stable theories of modules in 1980. In 1984 Shelah extended this result to arbitrary -stable theories.
The main objective of this talk is to show that Vaught's conjecture is true for any complete theory of modules over 1) countable serial rings; 2) countable commutative Prüfer rings; 3) countable hereditary noetherian prime rings and 4) tame hereditary finite-dimensional algebras over a countable infinite field. In all these cases we prove that, if has few types, then it is -stable, making it possible to apply Garavaglia's result.
We give a complete classification of modules with few models over countable serial rings, and of modules with few types over countable commutative Prüfer rings. A similar description holds true for modules over a string algebra over a finite or countable field .
We also discuss some partial results on Vaught's conjecture for modules over group rings and for modules over pullback rings.
Recently we defined the concept of right Baer modules over a ring . A module is called Baer if the right annihilator of any set of endomorphisms of , in is a direct summand of . This concept generalizes the concept of Baer rings. Kaplansky developed a theory of types for Baer rings, and Goodearl and Boyle extended it to nonsingular injective modules. We extend this type theory to Baer modules and, in particular, since nonsingular CS modules are Baer, to nonsingular CS modules (joint work with S.T. Rizvi).
A ring is called a right Harada ring if it is right Artinian and every non-small right -module contains a non-zero injective submodule. The first result in our report is the following Theorem: Let be a right perfect ring. Then is a right Harada ring if and only if every cyclic module is a direct sum of an injective module and a small module; if and only if every localmodule is either injective or small. We also prove that a ring is QF if and only if every cyclic module is a direct sum of a projective injective module and a small module; if and only if every local module is either projective injective or small. Finally, a right QF-3 right perfect ring is serial if and only if every right ideal is a direct sum of a projective module and a singular module.
For an algebra, consider the category of all pairs where is a finite length -module and a submodule of . Then is a full and exact subcategory of a module category, so every object in is a direct sum of indecomposables. It turns out that has Auslander-Reiten sequences, so we computed a formula for the Auslander-Reiten translation, which acts on the set of indecomposables in . Surprisingly, for a uniserial ring, this translation has period six. This is a report on joint work with Claus Michael Ringel.
(Joint work with Lianyond Xue)
Let be a Galois algebra over a commutative ring with Galois group . Then is a direct sum of Galois algebras such that each direct summand is a composition of a central Galois algebra and a commutative Galois algebra. A sufficient condition is also given under which is commutative.