on semi regularity and the total of rings and modules abstract let m and n be two modules over a ring r recent works by kasch schneider beidar mader and others have shown that some of the ring and module theory can be carried out in the context of hom r m n the study of substructures of hom r m n such as the radical the singular and co singular ideals and the total has raised new questions for research in this area this paper is a continuation of study of these substructures focusing on when the total is equal to the radical as well as their connections with semi regularity of hom r m n new results obtained include necessary and sufficient conditions for the total to equal the radical a description of the maximal regular sub bimodule of hom r m n the existence of the maximal semiregular ideal of a ring and answers to a number of existing open questions

endoproperties of modules and local duality abstract let r be any ring and n i i n i be a direct sum of finitely presented left r modules n i suppose that d n and d n i are the local duals of n and n i for each i i we prove that the lattice of endosubmodules of n is anti isomorphic to the lattices of matrix subgroups of d n and of m i i d n i as consequences n is endoartinian if and only if m or d n is endonoetherian and n is endonoetherian if and only if m or d n is pure injective we obtain in particular that if r is a krull schmidt ring and m is an indecomposable pure injective endonoetherian right r module which is the source of a left almost split morphism in mod r then m is endofinite as an application a ring r is of finite representation type if and only if every pure injective right r module is endonoetherian

classes of modules with the exchange property in searching for isomorphic refinements of direct sum decompositions of modules9 crawley and jonsson introduced an exchange property for modules that is strongly reminiscent of steinitz exchange lemma for vector spaces an r module m where r is an associative ring with identity is said to have the finite exchange property if m can be fitted into any finite sum grid as follows whenever a4 occurs as a direct summand of a finite direct sum a bielai then a m ci for suitable submodules ci of the ai 2 this concept led to very general extensions of the classical refinement theorems see e g 2 3 171 many of the problems suggested by crawley and jonsson in their pioneering paper are still open the most salient one does the finite exchange property imply the unrestricted exchange property we show that the answer is yes for modules which possess decompositions into indecomposable summands in the background of this observation stands the following

subsystems time scales and multimodeling through a couple of naive examples the control theorists are invited to re examine the role of modeling in the study of large scale dynamic systems instead of assuming the existence of n diagonally dominant blocks it is possible to justify one strongly coupled slow core and n weakly coupled fast subsystems this structure is exhibited with a physically meaningful choice of state variables the controls are introduced following the recent concept of multimodeling

robust output maneuvering for a class of nonlinear systems the output maneuvering problem involves two tasks the first called the geometric task is to force the system output to converge to a desired path parametrized by a continuous scalar variable q the second task called the dynamic task is to satisfy a desired dynamic behavior along the path this dynamic behavior is further specified via a time speed or acceleration assignment while the main concern is to satisfy the geometric task the dynamic task ensures that the system output follows the path with the desired speed a robust recursive design technique is developed for uncertain nonlinear plants in vectorial strict feedback form first the geometric part of the problem is solved then an update law is constructed that bridges the geometric design with the speed assignment the design procedure is illustrated through several examples

decomposition properties of strict mittag leffler modules we show that a not necessarily unitary ring with enough idempotents is left perfect if and only if there exists a cardinal number such that every flat strict mittag leffler module is a direct sum of generated modules several applications are given to the decomposition properties of modules into direct summands

coherency based decomposition and aggregation the relationship between coherency area decomposition and aggregation is investigated for a class of models of electromechanical networks including power system models it is established that coherency conditions and a physically meaningful area aggregate reproducing a prescribed set of system modes depend on external connections only coherency with respect to a set of slowest modes slow coherency and an earlier developed grouping algorithm are used to identify weakly coupled areas as demonstrated on a 48 machine model of the united states north east power system

every ℵ1 σ cs module is σ cs abstract it is proved that if m is a uniform module such that m 1 is cs then its endomorphism ring is local as a consequence it is shown that every 1 cs module is cs this solves open questions asked in contemp math 259 2000 467 j algebra 262 2003 194 on the way to these results it is also shown that the endomorphism ring of a uniform module m is local provided that every local summand of m 0 is a direct summand

graph weight allocation to meet laplacian spectral constraints we adjust the node and edge weightings of graphs using convex optimization to impose bounds on their laplacian spectra first we derive necessary and sufficient conditions that characterize the feasibility of spectral bounds given positive node and edge weightings synthesizing these conditions leads naturally to algorithms that exploit convexity to achieve several eigenvalue bounds simultaneously the algorithms we propose apply to many graph design problems as well as multi agent systems control finally we suggest efficient ways to accommodate larger graphs and show that dual formulations lead to substantial improvement in the size of graphs that can be addressed

model reduction and clusterization of large scale bidirectional networks this paper proposes two model reduction methods for large scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization first we present a krylov based model reduction method that guarantees a specified error precision in terms of the h norm positive tridiagonalization allows us to derive an approximation error bound for the input to state model reduction without computationally expensive operations such as matrix factorization second we propose a novel model reduction method that preserves network topology among clusters i e node sets in this approach we introduce the notion of cluster uncontrollability based on positive tridiagonalization and then derive its theoretical relation to the approximation error this error analysis enables us to construct clusters that can be aggregated with a small approximation error the efficiency of both methods is verified through numerical examples including a large scale complex network

modules with indecomposable decompositions that complement maximal direct summands generalizing a fundamental property of semisimple modules anderson w x and fuller 1 introduced the following important concept for direct decompositions of modules a decomposition m s m is said to ig i i complement direct summands in case for each direct summand a of m w there is a subset h i such that m s a m cf also 2 p ig h i x 141 such a decomposition is necessarily an indecomposable decomposition and it is an interesting problem to study under which conditions an indecomposable decomposition complements direct summands there is w x an extensive literature concerning this problem by 1 theorem 6 right perfect rings can be characterized by the property that all projective right modules have decompositions that complement direct summands fuller w x 7 showed that over a generalized uniserial ring every module has a decomposition that complements direct summands thus providing a first class of non semisimple rings satisfying this property more generally w x tachikawa 14 proved that every left and right module over a ring of finite representation type has a decomposition that complements direct summands and that the converse is also true was established by fuller and w x w x reiten 9 restricting just to one side fuller 8 showed that rings over which every right module has a decomposition that complements direct summands are precisely the rings over which every right module is a direct sum of finitely generated modules they are also called right pure semisim

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

stability theorems of stochastic difference equations abstract in this paper we prove the general comparison theorem for the difference inequalities and several stochastic stability theorems of the nonlinear difference equations using this theorem our theorems generalize ma and caughey s theorems

square free modules with the exchange property we prove that a square free module with finite exchange has full exchange more generally if r is an exchange ring with r j r abelian and r is endowed with a left linear hausdorff complete topology then r is a full exchange ring this provides an overarching framework for capturing many other results in the literature such as the fact that quasi continuous modules with finite exchange have full exchange we further show that square free modules with exchange satisfy an infinite version of the c3 property

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

continuous modules are clean ar ingr is said to be clean if every element of r is a sum of an idempotent and a unit the class of clean rings is quite large and includes for instance semiperfect rings and thus finite rings and rings of linear transformations of vector spaces we prove that the endomorphism ring of every continuous or discrete module is clean

stable range one for rings with many idempotents an associative ring r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit the purpose of this note is to prove the following facts theorem 3 an exchange ring r has stable range 1 if and only if every regular element of r is unit regular theorem 5 if r is a strongly 7r regular ring with the property that all powers of every regular element are regular then r has stable range 1 the latter generalizes a recent result of goodearl and menal 5 let r be an associative ring with identity r is said to have stable range 1 if for any a b e r satisfying ar br r there exists y e r such that a by is a unit this definition is left right symmetric by vaserstein 9 theorem 2 furthermore by a theorem of kaplansky all one sided units are two sided in rings having stable range 1 cf vaserstein 10 theorem 2 6 it is well known that a von neumann regular ring r has stable range 1 if and only if r is unit regular see for example goodearl 4 proposition 4 12 call a ring r strongly 7 regular if for every element a e r there exist a number n depending on a and an element x e r such that an an lx this is in fact a two sided condition 3 it is an open question whether all strongly 7r regular rings have stable range 1 goodearl and menal 5 proved that strongly 7r regular rings are unit regular and hence have stable range 1 theorem 5 8 p 278 in this note we first extend the above result for von neumann regular rings to a larger class of rings which includes all strongly 7r regular rings 7r regular rings von neumann regular rings and algebraic algebras as an application of this we prove that a strongly 7r regular ring r has stable range 1 if powers of every regular element are regular the latter is a generalization of the abovementioned result of goodearl and menal for strongly 7r regular regular rings as one can see from our proofs rings in these classes have a large supply of idempotents throughout r stands for an associative ring with identity and j r for the jacobson radical of r modules are unitary right r modules except otherwise specified for other undefined terms readers are referred to 4 let mr be a right r module following crawley and jonsson 2 mr is said to have the exchange property if for every module ar and any two received by the editors january 24 1994 and in revised form may 3 1994 originally communicated to the proceedings of the ams by k a goodearl 1991 mathematics subject classification primary 1 6d70 1 6p70

a novel reduced model for electrical networks with constant power loads we consider a network preserved model of power networks with proper algebraic constraints resulting from constant power loads both for the linear and the nonlinear differential algebraic model of the network we derive explicit reduced models which are fully expressed in terms of ordinary differential equations for deriving these reduced models we introduce the projected incidence matrix which yields a novel decomposition of the reduced laplacian matrix with the help of this new matrix we provide a complementary approach to kron reduction which is able to cope with constant power loads and nonlinear power flow equations

sigma cotorsion rings abstract it is proved that a ring r is right perfect if and only if it is cotorsion as a right module over itself several other conditions are shown to be equivalent for example that every pure submodule of a free right r module is strongly pure essential in a direct summand or that the countable direct sum c r 0 of the cotorsion envelope of r r is cotorsion if c r is a flat cotorsion module then c r admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring the jacobson radical j s of the endomorphism ring s end r c is characterized as the maximum ideal that acts locally t nilpotently on c r if r is semilocal and c c r then the radical consists of those endomorphisms f c c whose image is contained in cj

area aggregation and time scale modeling for sparse nonlinear networks model reduction and aggregation are of key importance for simulation and analysis of large scale systems such as molecular dynamics large swarms of robotic vehicles and animal aggregations we study a nonlinear network which exhibits areas of internally dense and externally sparse interconnections the densely connected nodes in these areas synchronize in the fast time scale and behave as aggregate nodes that dominate the slow dynamics of the network we first derive a singular perturbation model which makes this time scale separation explicit and next prove the validity of the reduced model approximation on the infinite time interval

a comparison theorem for difference inequalities in the study of growth and decay properties of nonlinear evolution equations in particular hyperbolic and parabolic partial differential equations as t the following difference inequality which is satisfied by an energy expression arises mathop sup limits t leq s leq t 1 u s 1 alpha leq c 1 t r u t u t 1 g t

on rings whose projective modules have the exchange property it is well known that all projective modules over a perfect ring have the exchange property this was shown by yamagata 32 and independently by harada and ishii lo provided that the identity of the ring can be expressed as a sum of orthogonal primitive idempotents the converse is also true 32 that it fails in general was demonstrated by kutami and oshiro ls who constructed a nonartinian boolean ring whose projective modules nevertheless have the exchange property as it turned out this outsider example is representative of an extensive class of rings in fact both perfect and regular rings are subsumed in the hypotheses of the following

radical endomorphisms of decomposable modules an element of the jacobson radical of the endomorphism ring of a decomposable module is characterized in terms of its action on the components of the decomposition this extends to arbitrary decomposable modules a result previously known only for the special case of free modules

a unified view of aggregation and coherency in networks and markov chains abstract this paper presents a unified treatment of aggregability lumpability coherency reversibility partial balance and similar properties appearing in different fields including power systems and queueing theory a coherency condition well known in power systems implies the existence of a finite state filter for markov chains while aggregability and coherency yield a new condition for decentralized computation of the invariant measure

half orthogonal sets of idempotents we improve several results in the literature focused on lifting idempotents by either removing the lifting hypothesis or weakening other assumptions we prove that countable sets of idempotents which are orthogonal modulo an enabling ideal lift to orthogonal idempotents left associates of liftable idempotents also lift modulo the jacobson radical additionally we exhibit situations when half orthogonal sets of idempotents can be orthogonalized by multiplying by a unit we finish by proving a number of results on directs sums of modules with the exchange property

weak krull schmidt for infinite direct sums of uniserial modules w x the well known krull schmidt azumaya theorem 2 says that if m s m s n are two indecomposable decompositions of a i j ig i jg j module m where m has local endomorphism ring for each i g i then i there is a bijection s i a j such that m n for every i g i in this i s i theorem the local endomorphism ring hypothesis is crucial to ensure the uniqueness of the decomposition of m into indecomposable summands and it is always tempting to ask whether under some weaker hypotheses the indecomposable decomposition of a module is unique this was done w x for example by anderson and fuller 1 theorem 12 4 who showed that the conclusion of the krull schmidt azumaya theorem still remains valid for modules with an indecomposable decomposition that complements maximal direct summands in general however there exist various examples in the literature which show that krull schmidt may fail even under rather good hypotheses on the rings or on the indecomposable sum w x w x w x mands see e g 5 7 and 8 in 1975 warfield proved that every finitely presented module over a serial ring is a finite direct sum of uniserial modules and asked whether

discrete inequalities of the gronwall bellman type in n independent variables ii soit r 0 n l ensemble des entiers non negatifs et n un entier positif soit u x k x n n r et f x s n 2n r avec s x si u x k x s 0 x 1 f x s u s pour x n n alors u x k x sd 1 0 xu 1 1 1 s 0 x 1 f x s 1 s ou k x sup k x 0 s x f x s sup f t s 0 t x

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods