clean rings some clean and almost clean baer rings and von neumann algebras a ring is clean resp almost clean if each of its elements is the sum of a unit resp regular element and an idempotent in this paper we define the analogous notion for rings a ring is clean resp almost clean if its every element is the sum of a unit resp regular element and a projection although clean is a stronger notion than clean for some rings we demonstrate that it is more natural to use the theorem on cleanness of unit regular rings from camillo and khurana 2001 4 is modified for cleanness of regular rings that are abelian or reduced or armendariz using this result it is shown that all finite type i baer rings that satisfy certain axioms considered in berberian 1972 2 and vas 2005 26 are almost clean in particular we obtain that all finite type i aw algebras thus all finite type i von neumann algebras as well are almost clean we also prove that for a baer ring satisfying the same axioms the following properties are equivalent regular unit regular left right morphic and left right quasi morphic if such a ring is finite and type i it is clean finally we present some examples related to group von neumann algebras and list some open problems

the large time stability of sound waves we demonstrate the existence of solutions to the full 3 3 system of compressible euler equations in one space dimension up to an arbitrary timet 0 in the case when the initial data has arbitrarily large total variation and sufficiently small supnorm the result applies to periodic solutions of the euler equations a nonlinear model for sound wave propagation in gas dynamics our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem this length scale plays no role in 2 2 systems or any system possessing a full set of riemann coordinates nor in the small total variation problem forn n systems the cases originally addressed by glimm in 1965 recent work by a number of authors has demonstrated that when the total variation is sufficiently large solutions of 3 3 systems of conservation laws can in general blow up in finite time independent of the supnorm due to amplifying instabilities created by the non trivial lie algebra of the vector fields that define the elementary waves for the large total variation problem there is an interaction between large scale effects that amplify and small scale effects that are stable and we show that the length scale on which this interaction occurs isd in the limitd we recover glimm s theorem and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp

idempotents in ring extensions abstract the aim of the paper is to study idempotents of ring extensions r s where s stands for one of the rings r x 1 x 2 x n r x 1 1 x 2 1 x n 1 r x 1 x 2 x n we give criteria for an idempotent of s to be conjugate to an idempotent of r using our criteria we show in particular that idempotents of the power series ring are conjugate to idempotents of the base ring and we apply this to give a new proof of the result of p m cohn 2003 4 theorem 7 that the ring of power series over a projective free ring is also projective free we also get a short proof of the more general fact that if the quotient ring r j of a ring r by its jacobson radical j is projective free then so is the ring r

rings with property a and their extensions abstract a commutative ring r has property a if every finitely generated ideal of r consisting entirely of zero divisors has a nonzero annihilator we continue in this paper the study of rings with property a we extend property a to noncommutative rings and study such rings moreover we study several extensions of rings with property a including matrix rings polynomial rings power series rings and classical quotient rings finally we characterize when the space of minimal prime ideals of rings with property a is compact

characterizing nilpotent leibniz algebras by a new bound on their second homologies abstract let l be a finite dimensional nilpotent leibniz algebra such that dim l n and dim l 2 m 0 in this paper we prove dim h l 2 l n m 2 n m m 2 where h l 2 l is the second leibniz homology of l as a consequence for a non abelian nilpotent leibniz algebra l we find that s l n 1 2 1 dim h l 2 l 0 furthermore we determine all finite dimensional nilpotent leibniz algebras with s l less than or equal to three

the armendariz property on ideals abstract in the present note we study the armendariz property on ideals of rings introducing a new concept which unifies the armendariz property and the insertion of factors property simply ifp for rings in relation with this work we investigate rings over which polynomial rings are ifp called strongly ifp rings which generalize both ideal armendariz rings and strongly reversible rings the classes of minimal noncommutative ideal armendariz rings and strongly ifp rings and the classes of minimal non abelian ideal armendariz rings and strongly ifp rings are completely determined up to isomorphism it is also shown that a local ring is armendariz symmetric and strongly reversible hence ideal armendariz when the cardinality of the jacobson radical is 4

a structure theorem for leibniz homology abstract presented is a structure theorem for the leibniz homology hl of an abelian extension of a simple real lie algebra g as applications results are stated for affine extensions of the classical lie algebras sl n r so n r and sp n r furthermore hl h is calculated when h is the lie algebra of the poincare group as well as the lie algebra of the affine lorentz group the structure theorem identifies all of these in terms of g invariants

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

properties which do not pass to classical rings of quotients abstract we construct examples of ore rings satisfying some standard ring theoretic properties for which the classical rings of quotients do not satisfy those properties examples of properties which do not pass to rings of quotients include abelian dedekind finite semicommutative 2 primal and ni in the process of constructing these examples we correct the literature we also introduce two important construction techniques

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

systems of conservation laws with invariant submanifolds systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation multicomponent chromatogra phy as well as in the study of nonlinear motion in elastic strings here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws this coinci dence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension in the case of 2 x 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide introduction systems of conservation laws which have coinciding shock and rarefaction curves arise in the study of oil reservoir simulation nonlinear wave motion in elastic strings as well as in multicomponent chromatography 1 4 5 6 9 11 12 these systems have many interesting features the riemann problem for these equations can be explicitly solved in the large and wave interactions have a simplified structure even in the presence of a nonconvex flux function for this reason these systems represent some of the few examples for which the cauchy problem has been solved for arbitrary data of bounded variation also hyperbolic degeneracies appear in each of these systems in the present paper we are concerned with characterizing the phenomenon of coinciding shock and rarefaction curves this phenomenon turns out to be a special case of the general phenomenon of a submanifold of r being invariant for weak solutions of a given system of n conservation laws invariant manifolds of higher dimension also appear in physical conservation equations in which the conserved quantities represent concentrations i e when the concentrations c of any of the species vanish the system must reduce to a lower order system that expresses the conservation of the remaining species thus c 0 defines a manifold which is invariant in the sense that solutions that start on the manifold remain there for all time moreover such an invariant manifold of dimension n 1 is the boundary of an dimensional invariant region in the sense of chueh conley and smoller 14 in the case of

a taxonomy of 2 primal rings abstract various conditions on a noncommutative ring imply that it is 2 primal i e the ring s prime radical coincides with the set of nilpotent elements of the ring we will examine several such conditions and show that their known interdependencies are their only ones of particular interest will be the ps i condition on a ring i e every factor ring modulo the right annihilator of a principal right ideal is 2 primal we will see that even within a fairly narrow class of rings ps i is a strictly stronger condition than 2 primal we will show that the ps i condition is left right asymmetric we will also study the interplay between various types of semilocal rings and various types of 2 primal rings the kothe conjecture will make a cameo appearance in section 6 we will examine subideals of prime ideals of commutative rings that are invariant under derivations

global entropy solutions to weakly nonlinear gas dynamics abstract entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics through a modified glimm scheme an approximate solution sequence is constructed and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws which gives not only the existence and uniqueness but also the uniform total variation bounds for the entropy solutions

generalized weak armendariz rings and their extensions for an endomorphism of a ring r we introduce the concepts of weak ar mendariz rings and weak quasi armendariz rings to investigate the relative weak armendariz properties relative to in a general setting the two new classes of rings extend the classes of weak armendariz rings and weak quasi armendariz rings respectively and thus we can study various weak armendariz properties of a ring from a new perspective characterizations of weak armendariz rings and weak quasi armendariz rings are given and some examples and counter examples needed in the process are also constructed some known results are unified and generalized keywords weak armendariz rings weak quasi armendariz rings weak armendariz rings weak quasi armendariz rings weak mccoy modules

nilpotent elements and armendariz rings we study the structure of the set of nilpotent elements in armendariz rings and introduce nil armendariz as a generalization we also provide some new examples by proving that if d is a k algebra and n 2 the coproduct d kk x xn 0 is armendariz if and only if d is a domain with k 0 as its group of units finally we study the conditions under which the polynomial ring over a nil armendariz ring is nil armendariz which is related to a question of amitsur

semi commutativity and the mccoy condition abstract we prove that all reversible rings are mccoy generalizing the fact that both commutative and reduced rings are mccoy we then give an example of a semi commutative ring that is not right mccoy at the same time we also show that semi commutative rings do have a property close to the mccoy condition

leibniz algebras with small derived ideal abstract a nilpotent leibniz algebra l is called extra special if dim z l dim l 2 1 in this paper the structure the second homology and the capability of extra special leibniz algebras are completely determined

decay of almost periodic solutions of conservation laws we consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables which are almost periodic in the space variables in a generalized sense introduced by stepanoff and wiener which extends the original one of h bohr we prove that if u x t is such a solution whose inclusion intervals at time t with respect to 0 satisfy l epsiv t t 0 as t and such that the scaling sequence u t x t u t x t t is pre compact as t in l loc 1 d 1 then u x t decays to its mean value which is independent of t as t the decay considered here is in l 1 loc of the variable x t which implies as we show that as t where m x denotes taking the mean value with respect to x in many cases we show that if the initial data are almost periodic in the generalized sense then so also are the solutions we also show in these cases how to reduce the condition on the growth of the inclusion intervals l t with t as t for fixed 0 to a condition on the growth of l 0 with as 0 which amounts to imposing restrictions only on the initial data we show with a simple example the existence of almost periodic non periodic functions whose inclusion intervals satisfy any prescribed growth condition as 0 the applications given here include inviscid and viscous scalar conservation laws in several space variables some inviscid systems in chromatography and isentropic gas dynamics as well as many viscous 2 2 systems such as those of nonlinear elasticity and eulerian isentropic gas dynamics with artificial viscosity among others in the case of the inviscid scalar equations and chromatography systems the class of initial data for which decay results are proved includes in particular the l generalized limit periodic functions our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws

long time existence of entropy solutions to the one dimensional non isentropic euler equations with periodic initial data the non isentropic euler system with periodic initial data in mathbb r 1 is studied by analyzing wave interactions in a framework of specially chosen riemann invariants generalizing glimm s functionals and applying the method of approximate conservation laws and approximate characteristics an mathcal o varepsilon 2 lower bound is established for the life span of the entropy solutions with initial data that possess varepsilon variation in each period

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the