on the ideals of a noetherian ring we construct various examples of noetherian rings with peculiar ideal structure for example there exists a noetherian domain r with a minimal nonzero ideal 1 such that r i is a commutative polynomial ring in n variables and a noetherian domain s with a second layer clique that is not locally finite the key step in the construction of these rings is to idealize at a right ideal i in a noetherian domain t such that t l is not artinian introduction the aim of this paper is to construct several examples concerned with the ideal structure of a noetherian ring we will delay discussion of their relevance until we are ready to construct them in 2 3 and 4 and simply describe their properties at this stage however for the most part these examples complement the positive results in the author s survey article 12 example 1 see theorem 2 3 let m be a positive integer there exists a noetherian domain r with a unique minimal nonzero ideal i such that r i is a polynomial ring in m variables in particular r has infinitely many prime ideals example 2 theorem 3 2 there exists a prime noetherian ring r with goldie rank two written grk r 2 yet for each n c n there exists a prime ideal qn of r with grk r qn 2n 1 further nqn 0 example 3 theorem 4 1 there exists a noetherian ring r and two prime ideals p1 and p2 of r such that there exists an ideal link between p1 and p2 but such that pi and p2 do not belong to the same second layer clique example 4 theorem 4 4 there exists a noetherian domain r and a second layer clique of prime ideals of r that is not locally finite this clique also satisfies the second layer condition the connecting theme between these examples and the crucial step in their construction is the concept of the idealizer i i f e s fi c i of a right ideal i in a noetherian ring s of course since robson s seminal paper 8 this has been a well known and productive method of producing interesting examples however in previous examples s i has been artinian unfortunately this only produces finitely many new ideals whereas the present examples are concerned with the properties of an infinite set of prime ideals thus we are forced to idealize at a right ideal i such that s i is not artinian in general such an idealizer will not be left or right received by the editors june 11 1984 and in revised form july 30 1984 1980 mathematics subject classification primary 16a33 16a66 secondary 16a08

square lattice random potts model criticality and pitchfork bifurcation abstract within a real space renormalization group framework based on self dual clusters we discuss the criticality of the quenched bond mixed q state potts ferromagnet on a square lattice on qualitative grounds we exhibit that crossover from the pure fixed point to the random one occurs while q increases through a pitchfork bifurcation the relationship with the harris criterion is analyzed on quantitative grounds we present high precision numerical values for the critical temperatures corresponding to various concentrations of the coupling constants j 1 and j 2 and various ratios j 1 j 2 the pure random and crossover critical exponents are discussed as well

prime ideals in group algebras oe polycyclic by finite groups index such that the prime ideals of k g0 ca n de described in a particularly nice manner furthermore he showed that this type of description does not in general apply to k g in this paper we offer a slightly different somewhat more complicated formulation which does indeed describe the primes of k g observe that the group algebra k g can be written as k g k g0 g go a crossed product of the finite group h g go over the ring r k g0 furthermore a recent paper 5 of the authors studies the general crossed product situation r h with h finite and offers a rather complicated scheme for describing the prime ideals of

a nonembeddable noetherian ring abstract we provide an example of a left and right noetherian c algebra which cannot be embedded into any artinian ring

the potts model and flows iii standard and subgraph break collapse methods for pt ii see ibid vol 19 p 1655 1986 an algorithm is developed for the exact calculation of the many spin correlation functions of potts model clusters which is more efficient than the standard break collapse method successfully used in real space renormalisation group calculations the improved performance is based on a relationship which at any stage of the calculation allows the replacement of certain subgraphs by single effective edges thereby decreasing the number of iterations needed the authors method avoids as in the standard one the time consuming summation over spin states and may be used to extend series expansion and real space renormalisation group calculations on crystal lattices both methods are based on a number of combinatorial formulae the proofs of which are given in this paper

cancellation over commutative rings of dimension one and two all rings in this paper are commutative and noetherian and all modules are finitely generated we will study the question if a c z b c is a z b here a b and c are modules over a ring r in 1962 chase 4 gave an affirmative answer for r k x y the polynomial ring in two variables over an algebraically closed field k provided a and b are torsionfree and char k trank a in the first section of this paper we extend chase s result to include all two dimensional regular affine k domains most of the paper deals with one dimensional rings in section 2 we prove some general results which we hope will eventally lead to some sort of structure theory for torsionfree modules over one dimensional reduced rings with finite normalization in this context we show that one can cancel from projectives that is the answer is yes when a and b are projective if and only if pit r pit z where g is the normalization when a and b are assumed to be merely torsionfree the problem is more subtle but we have some partial results the last three sections deal with three naturally occurring examples of one dimensional rings coordinate rings of curves quadratic orders and integral group rings of finite abelian groups in each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules many of the ideas in this paper are the direct or indirect results of innumerable conversations with raymond heitmann and lawrence levy in particular levy discovered a serious error in the proof of the main theorem of an earlier version and heitmann showed by example that the statement itself was incorrect i am extremely grateful to both of them for their interest and insight

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

the potts model and flows ii many spin correlation function for pt i see ibid vol 19 p 411 1985 a graph theoretic analysis is made of the m spin correlation functions of the lambda state potts model in paper i the correlation functions for m 2 were expressed in terms of rooted mod lambda flow polynomials the authors introduce a more general type of polynomial the partitioned m rooted flow polynomial which plays a fundamental role in the calculation of the multispin correlation functions the m rooted equivalent transmissivities of tsallis and levy 1981 are interpreted in terms of percolation theory and are expressed as linear combinations of the above correlation functions

multiplicity and concentration of solutions for a quasilinear choquard equation in this paper we study a quasilinear choquard equation involving the p laplacian operator and a potential function v under suitable assumptions on v and the nonlinearity we prove the existence multiplicity and concentration of solutions for the equation by variational methods

idealizer rings and noncommutative projective geometry abstract we study noetherian graded idealizer rings which have very different behavior on the right and left sides in particular we construct noetherian graded algebras t over an algebraically closed field k with the following properties t is left but not right strongly noetherian t kt is left but not right noetherian and t ktop is noetherian the left noncommutative projective scheme t proj is different from the right noncommutative projective scheme proj t and t satisfies left d for some d 2 yet fails right 1

k theory and right ideal class groups for hnp rings let r be an hereditary noetherian prime ring let s be a dedekind closure of r and let r be the category of finitely generated s torsion r modules it is shown that for all i 0 there is an exact sequence 0 ki t ki r ki s o if i 0 or r has finitely many idempotent ideals then this sequence splits a notion of right ideal class group is then introduced for hereditary noetherian prime rings which generalizes the standard definition of class group for hereditary orders over dedekind domains it is shown that there is a decomposition o r cl r ffl f where f is a ree abelian group whose ra k depends on the number of idempotent maximal ideals of r moreover there is a natural is morphism cl r cl s and this decomposition corresponds closely to the splitting of the above exact sequence for ko

generating modules efficiently algebraic k theory for noncommutative noetherian rings abstract the well known forster swan theorem and the stable version of this result given by eisenbud and evans give a bound on the number of generators of a module over a commutative ring in terms of local data in this paper we generalise these results to arbitrary noetherian rings as an application we prove versions of serre s theorem and the cancellation theorem over such rings

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

ground states and non existence results for choquard type equations with lower critical exponent and indefinite potentials abstract in this paper we study following choquard equation with lower critical exponent u v x u i u n 1 u n 1 u f x u x r n u h 1 r n where n 1 i is the riesz potential and v r n r allows to be sign changing under some mild assumptions imposed on the nonlinearity f we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case respectively the characterization of the ground states is also investigated by a direct approach that are constrained to the nehari manifold moreover we show a non existence result for the equation via a generalized poho aev identity established for the non autonomous nonlinearity f the results extend and improve some recent ones in the literature

control of the immune response by a threshold automata model on a lattice abstract the kut mean field model for the immune response is generalized on a lattice using boolcan and threshold automata it is shown that the lattice model can exhibit properties that the mean field model cannot even if the biological meaning of the lattice geometry is dubious the attractors of the mean field model become unstable under mixing of populations and the system may reach the paralysis state instead of the memory vaccinated the probability of vaccination is calculated analytically and checked numerically a threshold automata model which includes interactions with the antigen and uses random neighbours and random time delays is proposed allowing control of the immune response by the concentration of antigen finally quantitative results of the dynamics are given using the model itself unlike previous works that used differential equations the numerical results show low and high dose paralysis versus medium dose vaccination

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

criticality of the discrete n vector ferromagnet in planar self dual lattices the authors study an extended version of the discrete n vector or cubic ferromagnetic model within a real space renormalisation group approach which preserves the two spin correlation function the n evolution for real values of n of the wheatstone bridge hierarchical lattice phase diagram which presents paramagnetic intermediate nematic like and ferromagnetic phases as well as of the thermal nu and crossover phi critical exponents is presented the self avoiding walk problem is recovered in the n to o limit and the so called corner rule is re obtained in a larger context the ising n and 2n state potts ferromagnets are recovered as particular cases an interchange of stability occurs at n n approximately 6 9 in such a way that the 2n state potts special point where all three existing phases join is multicritical if n n but only critical if n n consistently phi n o for the cubic model nu n presents a maximum at n nmax approximately 1 5 the results are exact for all n for the wheatstone bridge hierarchical lattice and approximate for n or 2 for the square lattice last but not least they discuss the connection between the present approach and the phenomenological renormalisation group

the influence of ideal structure on representation theory the purpose of this paper is to explore the representation theoretic consequences of the two sided ideal structure of a noetherian ring with special emphasis on an additional combinatorial structure on the prime spectrum of the ring the graph of links we obtain existence theorems for extensions and for rings satisfying the strong second layer condition we obtain extensive information about the indecomposable injective modules particularly detailed results are obtained when the ring is polynormal

z λ model and flows subgraph break collapse method a recursive algorithm previously developed for carrying out renormalisation calculations for the lambda state potts model is generalised to the z model the relations used are based on an expression the authors derive for the pair correlation function in terms of mod lambda flows which represents an extension of a similar result for the partition function previously obtained by biggs 1976 the use of flows enables them to prove and extend the formulae which appear in the break collapse method of mariz and co workers 1989 it is argued that the use of fixed flow bonds rather than the precollapsed bonds used by the latter authors leads to a more efficient algorithm

non holonomic modules over weyl algebras and enveloping algebras a number of counterexamples in the theory of enveloping algebras and noetherian rings are presented in particular we construct non holonomic simple modules over the weyl algebra a and the enveloping algebra u s1 z xs12 further we show that u s1 z x s12 is not weakly ideal invariant and that there exist simple s1 z x s12 modules m and e with e finite dimensional such that e is not artinian if m is a module over a ring r then the gelfand kirillov dimension of m written gk dim m is defined in 7 suppose that m is a simple module over a noetherian ring r and that g k d i m r oc then m is called holonomic if g k d i m m 8 9 the reason for the definition is that in the case when r is a weyl algebra or the enveloping algebra of a finite dimensional algebraic lie algebra holonomic modules have particularly nice properties

uniqueness of ground states of some coupled nonlinear schrodinger systems and their application abstract we establish the uniqueness of ground states of some coupled nonlinear schrodinger systems in the whole space we firstly use schwartz symmetrization to obtain the existence of ground states for a more general case to prove the uniqueness of ground states we use the radial symmetry of the ground states to transform the systems into an ordinary differential system and then we use the integral forms of the system more interestingly as an application of our uniqueness results we derive a sharp vector valued gagliardo nirenberg inequality

hereditary noetherian prime rings 1 integrality and simple modules this is the first of three papers that aim to bring the known theory of projective modules over a hereditary noetherian prime ring r up to roughly the same level as the well known commutative case where r is a dedekind domain this first paper lays the foundations by introducing the notion of an integral extension s of r in the goldie quotient ring of r and elucidating the relationship between integrality and the r module structure of simple s modules

thermal transmissivity in discrete spin systems formulation and applications thermal transmissivity a notion which arises empirically and leads to considerable simplification of analyses of discrete spin systems is of fundamental interest in graphical and real space renormalization group approaches the authors present a formulation of the theory of thermal transmissivity they show that the situation is different depending on whether the interaction matrices commute in the commuting case the thermal transmissivity is given by the eigenvalues of the interaction matrix while in the non commuting case it is given by the eigenvalues as well as matrices which block diagonalize the interaction matrix the meaning of the thermal transmissivity in diagrammatic analyses of spin models is also elucidated and their results are illustrated by examples finally they present a general formulation of disorder solutions for spin models in terms of thermal transmissivities

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

wild torsion modules over weyl algebras and general torsion modules over hnps let a 1 be the first weyl algebra over a field f of characteristic zero we show that the category of a 1 modules of finite length is wild despite the fact that every a 1 module of finite length is cyclic in fact the category of a 1 modules not necessarily finitely generated of socle height 2 is wild in a very strong sense among the applications we show i essentially any f algebra can occur as the endomorphism algebra of an a 1 module of socle height 2 ii a 1 has very large indecomposable modules of finite length iii there is an hnp hereditary noetherian prime ring that has indecomposable modules of finite length requiring arbitrary many generators we also complete the basic theory of finitely generated modules over general hnps by showing that every such module is a direct sum of right ideals and homomorphic images of right ideals and by proving a simultaneous decomposition theorem for an arbitrary projective module and submodule

non finitely generated projective modules over generalized weyl algebras abstract we classify infinitely generated projective modules over generalized weyl algebras for instance we prove that over such algebras every projective module is a direct sum of finitely generated modules

nonexistence and optimal decay of supersolutions to choquard equations in exterior domains we consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the riesz potential of a power this family of equations includes the choquard or nonlinear schrodinger newton equation we show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains the same techniques yield optimal decay rates when supersolutions exist 2013 elsevier inc

state spaces finite algebras and skew group rings k goodearl gl considered the state space st r of a noetherian ring r and in joint work with r b warfield gw described the structure of the sate space of commutative rings hereditary noetherian prime rings and orders over dedekind domains in this work we give some properties of the state space of a ring in relation to the ideas of localization on noncommutative rings and of ramification in generically galois actions we show how the structure of st r depends on the property of localization at prime ideals some general results are given for the state space of a finite algebra over a commutative noetherian ring then we apply these results to st s g where s is a commutative noetherian domain g is a finite group acting faithfully as automorphisms of the ring s and 1 gi is invertible in s wc show that the state space st s g is the affine continuous image of an amalgamation of the state spaces of the ring localized at non trivial clans in particular that those arising from the ramified primes of s suffice we show that st s g is a singleton if and only if the action is g galois on s finally for s a dedekind domain we give a structure theorem for the state space of the skew group ring s g

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

affiliated series and extensions of modules the main purpose of this paper is to examine the influence on the representation theory of a noetherian ring of the prime ideal structure of the ring the results concern the affiliated primes of a module defined below we show how the affiliated primes are connected with the annihilator primes and we prove a uniqueness theorem for affiliated primes under certain conditions we then proceed to analyse what affiliated primes can appear in terms of the ideal structure of the ring with special reference to the fundamental primes of injective modules our starting point is a theorem of jategaonkar which analyses the structure of modules with exactly two affiliated primes and which already has consequences for uniserial modules of length two we give a generalization of this result for modules with longer affiliated series and in particular for uniserial modules of length greater than two in particular we show that some of the possibilities one might have envisioned in the light of jategaonkar s theorem cannot in fact occur and we demonstate a somewhat unexpected connection between affiliated primes and annihilator primes to put all this material in context we begin section one with an analysis of what it all means for uniserial modules of length three we then turn to the uniqueness question for affiliated primes here and in the rest of the paper we find it desirable to have some kind of dimension 164 0021 8693 91 3 00

on the multiplicity and concentration of positive solutions for a p fractional choquard equation in rn abstract in this paper we deal with the following fractional choquard equation e s p p s u v x u p 2 u e n 1 x f u f u in r n u w s p r n u 0 in r n where e 0 is a small parameter s 0 1 p 1 n s p p s is the fractional p laplacian v is a positive continuous potential 0 s p and f is a continuous superlinear function with subcritical growth using minimax arguments and the ljusternik schnirelmann category theory we obtain the existence multiplicity and concentration of positive solutions for e 0 small enough

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

symmetric properties of positive solutions for fully nonlinear nonlocal system abstract in this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain bounded or unbounded in r n via using a direct method of moving planes this extends the results in wang and niu 2017 and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order

geometric algebras on projective surfaces abstract let x be a projective surface let aut x and let l be a ample invertible sheaf on x we study the properties of a family of subrings parameterized by geometric data of the twisted homogeneous coordinate ring b x l in particular we find necessary and sufficient conditions for these subrings to be noetherian we also study their homological properties their associated noncommutative projective schemes and when they are maximal orders in the process we produce new examples of maximal orders these are graded and have the property that no veronese subring is generated in degree 1 our results are used in the companion paper s j sierra 2009 sie09 to give defining data for a large class of noncommutative projective surfaces

liouville theorems involving the fractional laplacian on a half space abstract let r n be the upper half euclidean space and let be any real number between 0 and 2 consider the following dirichlet problem involving the fractional laplacian 1 2 u u p x r n u 0 x r n instead of using the conventional extension method of caffarelli and silvestre 8 we employ a new and direct approach by studying an equivalent integral equation 2 u x r n g x y u p y d y applying the method of moving planes in integral forms we prove the non existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions

z 6 model criticality and break collapse method abstract the phase diagram and thermal and crossover exponents of the z 6 ferromagnet on the square lattice are calculated within a real space renormalization group rg scheme the obtained phase diagram exhibiting 4 phases corresponding to z 6 z 3 z 2 and completely broken symmetries contains all exactly known critical points and possibly is an excellent approximation everywhere except in the high temperature region where a soft phase is expected to appear however all the obtained results are exact in the wheatstone bridge hierarchical lattice in addition to these results we present an operational procedure the break collapse method which considerably simplifies the exact calculation of two spin correlation functions for arbitrary z 6 two rooted clusters frequent in rg approaches

the thomas fermi von weizsacker theory of atoms and molecules we place the thomas fermi von weizsacker model of atoms on a firm mathematical footing we prove existence and uniqueness of solutions of the thomas fermi von weizsacker equation as well as the fact that they minimize the thomas fermi von weizsacker energy functional moreover we prove the existence of binding for two very dissimilar atoms in the frame of this model

presentations over hnp rings with enough invertible ideals and torsionfree cancellation over neoclassical orders abstract we prove a uniqueness theorem for presentations of modules over a class of hereditary noetherian prime rings that includes the hereditary orders studied in integral representation theory the result also generalizes the elementary divisor theorem for non commutative pids a key part of the proof is an extension of a direct sum cancellation theorem of drozd to torsionfree modules over a class of non commutative rings that need not be module finite over their center

hereditary noetherian prime rings 2 finitely generated projective modules abstract let r be a hereditary noetherian prime ring we determine a full set of invariants for the isomorphism class of any finitely generated projective r module of uniform dimension at least 2 in particular we prove that p x q x implies p q whenever p has uniform dimension at least 2 among the applications of these results are necessary and sufficient conditions for the existence of a bound to the number of generators needed for right ideals of r

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

the potts model and flows i the pair correlation function it is shown that the partition function for the a state potts model with pair interactions is related to the expected number of integer mod a flows in a percolation model the relation is generalised to the pair correlation function the resulting high temperature expansion coefficients are shown to be the flow polynomials of graph theory we also prove an observation of tsallis and levy concerning the equivalent transmissivity of a cluster

nonexistence of positive solutions for an indefinite fractional p laplacian equation abstract in this paper we consider the following fractional p laplacian equation with indefinite nonlinearities p s u x x 1 u q x x r n where 0 s 1 p q 1 2 we use the direct method of moving planes to prove the solution of above fractional p laplacian equation is monotone increasing along x 1 direction then obtain that the above equation possess no positive bounded solution on the whole space there is no decay condition at infinity on solution for the fractional p laplacian and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p laplacian here we also cannot find a suitable auxiliary function for this fractional p laplacian to circumvent these difficulties we use the following two estimate to study the property of solution i we estimate the singular integrals defining the fractional p laplacian along a sequence of approximate maxima ii we estimate the lower and upper bound of the fractional p laplacian equation

idealisers in skew group rings abstract let c be a commutative noetherian domain g be a finitely generated abelian group which acts on c and b c g be the skew group ring for a prime ideal i c we study the largest subring of b in which the right ideal ib becomes a two sided ideal the idealiser subring we obtain necessary and sufficient conditions for when this idealiser subring is left and right noetherian we also give an example of these conditions in practice which translates to an interesting number theoretic problem

noncommutative graded domains with quadratic growth letk be an algebraically closed field and letr be a finitely generated connected gradedk algebra which is a domain of gelfand kirillov dimension two write the graded quotient ringq r ofr asd z z 1 for some automorphism of the division ringd we prove thatd is a finitely generated field extension ofk of transcendence degree one moreover we describer in terms of geometric data ifr is generated in degree one then up to a finite dimensional vector space r is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curvey this implies in particular thatr is noetherian thatr is primitive when and thatr is a finite module over its centre when ifr is not generated in degree one thenr will still be noetherian and primitive if has infinite order butr need not be noetherian when has finite order

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

additive rank functions in noetherian rings let r be a ring with unit element a function 1 that assigns to each finitely generated r module m a nonnegative integer 1 m such that j m a n 1 l for each short exact sequence 0 n f m l f 0 of finitely right r modules is called an additive rank function such a function is constant on isomorphism classes of finitely generated right r modules so that 1 m cr m 0 whenever ci a4 m is an injective endomorphism of m when r is semisimple artinian then an additive rank function arises very naturally by defining a m length m the number of simple direct summands in a decomposition of m in 1964 goldie 2 adapted this idea to finitely generated modules over a right noetherian ring the resulting additive rank function is presently called the reduced rank the concept lay dormant for over 10 years until goldie started using it in 1976 providing new and strikingly elegant proofs of several important results of ring theory some of these appear in 3 others are included in 11 since then the reduced rank has become a widely used tool particularly in the theory of noncommutative noetherian rings we recall the definition of the reduced rank p m of a finitely generated right r module m over a right noetherian ring r if n denotes the nilpotent radical of r then r n has a classical semisimple artinian quotient ring q r n for a finitely generated right r n module m set prin m length m q r n since r nq r n is flat prin is an additive rank function for finitely generated right r n modules for a finitely generated right r module m set

basic element theory of noncommutative noetherian rings abstract in this paper we propose a notion of basic element for noncommutative noetherian rings and derive the corresponding basic element theorem as in d eisenbud and e g evans jr generating modules efficiently theorems from algebraic k theory j algebra 27 1973 278 305 as corollaries to this theorem we obtain a number of results of stafford such as his generalizations of serre s splitting off theorem and bass s cancellation theorem

when every projective module is a direct sum of finitely generated modules we characterize rings over which every projective module is a direct sum of finitely generated modules and give various examples of rings with and without this property

embedding noetherian rings in artinian rings abstract a criterion is presented which is easy to check in specific examples and provides information as to when a given noetherian ring is not embeddable in an artinian one on the other hand we provide sufficient conditions for artinian embeddability of several types of noetherian rings that have no nontrivial right ideals of reduced rank zero

solutions of fully nonlinear nonlocal systems abstract in this paper we consider the system involving fully nonlinear nonlocal operators f u x c n p v r n g u x u y x y n d y f v x f v x c n p v r n g v x v y x y n d y g u x for carrying on the method of moving planes a narrow region principle and a decay at infinity are established then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space furthermore non existence of positive solutions to the nonlinear system on a half space is derived

indefinite fractional elliptic problem and liouville theorems abstract in this paper we consider an indefinite fractional elliptic problem a corresponding liouville type theorem is established furthermore we obtain a priori estimate for solutions in a bounded domain by blowing up and rescaling we also classify the solutions of some degenerate elliptic equation originated from the fractional laplacian

the n component cubic model and flows subgraph break collapse method we specialize to then component cubic model the subgraph break collapse method which we recently developed for thez model the cubic model has less symmetry than the potts model for which the method was originally developed but nevertheless it is still possible to reduce considerably the computational complexity of the generalz model our recursive algorithm is similar forn 2 to the break collapse method for thez 4 model proposed by mariz and co workers it allows the exact calculation for the partition function and correlation functions forn component cubic clusters withn as a variable without the need to examine all of the spin configurations an important application is therefore in real space renormalization group calculations

geometric idealizer rings let b b x be the twisted homogeneous coordinate ring of a projective variety x over an algebraically closed field k we construct and investigate a large class of interesting and highly noncommutative noetherian subrings of b specifically let z be a closed subscheme of x and let i b be the corresponding right ideal of b we study the subalgebra r k i of b under mild conditions on z and aut k x r is the idealizer of i in b the maximal subring of b in which i is a two sided ideal our main result gives geometric conditions on z and that determine the algebraic properties of r we say that n z is critically transverse if for any closed subscheme y of z for n 0 the sub schemes y and n z are homologically transverse we show that if n z is critically transverse then r is left and right noetherian has finite left and right cohomological dimension is strongly right noetherian but not strongly left noetherian and satisfies right d where d codim z but fails left 1 this generalizes results of rogalski in the case that z is a point in d we also give an example of a right noetherian ring with infinite right cohomological dimension partially answering a question of stafford and van den bergh further we study the geometry of critical transversality and show that it is often generic behavior in a sense that we make precise

positive and sign changing solutions to a nonlinear choquard equation abstract we consider the problem u w x u 1 x u p u p 2 u u h 0 1 where is an exterior domain in r n n 3 0 n p 2 2 n n 2 w c 0 r n inf r n w 0 and w x v 0 as x under symmetry assumptions on and w which allow finite symmetries and some assumptions on the decay of w at infinity we establish the existence of a positive solution and multiple sign changing solutions to this problem having small energy

anisotropic cubic lattice potts ferromagnet renormalisation group treatment the choice of a convenient self dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q state potts ferromagnet criticality the exact critical frontier and dimensionality crossover exponent hf as well as the expected universality behaviour renormalisation flow sense are recovered for any linear scaling factor b and all values of q q or 4 the b 2 and b 3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den nijs conjecture the same calculation is performed for all values of b for the exponent nu d 1 associated with the one dimensional limit and the exact result nu d 1 1 is recovered in the limit b to infinity