分次环的有限正规扩张

本文研究了群分次环的有限正规分次扩张问题．利用经典环论方法，得到一个群分次环与其有限正规分次扩张环之间关于分次Jacobson根和分次素根的关系，同时，给出了分次情形的Cutting down定理和Lying over定理．     

Homogeneous nilradicals over semigroup graded rings

In this paper we study the homogeneity of radicals defined by nilpotence or primality conditions, in rings graded by a semigroup S. When S is a unique product semigroup, we show that the right (and left) strongly prime and uniformly strongly prime radicals are homogeneous, and an even stronger result holds for the generalized nilradical. We further prove that rings graded by torsion-free, nilpotent groups have homogeneous upper nilradical. We conclude by showing that non-semiprime rings graded by a large class of semigroups must always contain nonzero homogeneous nilpotent ideals.     

Prime ideals of graded rings and related matters

Let R be a ring graded by an abelian group.We study prime ideals of R that are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients of R are investigated.The results are applied to study when R is a Jacobson ring in case R is a Z-graded ring or a group ring of a finitely generated abelian group, or in case R is right Noetherian and strongly graded by a polycyclic-by-finite group.     

Diagonalization of matrices over graded principal ideal domains

The main result of this paper is the analogue of the classical diagonal reduction of matrices over PIDs, for graded principal ideal domains. A method for diagonalizing graded matrices over a graded principal ideal domain is obtained. In Section 2 we emphasis on some applications. A procedure is given to decide whether or not a matrix defined over an ordinary Dedekind domain (i.e. nongraded), with cyclic class group, is diagonalizable. In case the answer is positive the diagonal form can be calculated. This can be done by taking a suitable graded PID which has the Dedekind domain as its part of degree zero. It turns out that, even in the case where diagonalization of a matrix over the part of degree zero is not possible, the diagonal representation over the graded ring contains useful information. The main reason for this is that the graded ring hasn't essentially more units than its part of degree zero. We illustrate this by considering the problem of von Neumann regularity of a matrix over a Gr-PID and to matrices over Dedekind domains with cyclic class group. These problems were the original motivation for studying diagonalization over graded rings.     

Radicals of algebras graded by cancellative linear semigroups

We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.

     

Free vibration analysis of discrete-continuous functionally graded circular plate via the Neumann series method

The Neumann series method has been used for the first time to solve the boundary value problem of free axisymmetric and nonaxisymmetric vibrations of continuous and discrete-continuous functionally graded circular plate on the basis of the classical plate theory. The equation of motion and the general solution for a functionally graded circular plate with a very complex system of a discrete elements attached, such as concentric ring masses, elastic supports, rotational springs, and damping elements are presented for the first time. The particular continuous solutions to the defined differential equations are obtained as the Neumann power series rapidly, absolutely, and uniformly convergent to the exact eigenfrequencies for any physically justified values of the plate's parameters on the basis of the properties of the obtained closed-form kernels of the Volterra integral equations. The multiparametric nonlinear characteristic equations for plate with classical and nonclassical boundary conditions are defined and numerically solved to obtain the full spectrum of eigenfrequencies in a simple way. The effects of the position and stiffness of ring supports and of singularities as the radii of supports shrink to the center of the plate on the dimensionless eigenfrequencies of homogeneous and functionally graded circular plate with sliding support and elastic constraints are comprehensively studied and presented for the first time. The accuracy of the proposed low-computational-cost method is demonstrated by comparison of the numerical results with those available in the literature.     

弱分次周期根

对一般群分次环境立了弱分次周期根的概念,证明它是一个分次特殊根,给出了它的分次模刻,讲座了它与自反周期根的关系。     

Prime graded modules

In this paper, properties of prime and strongly prime graded modules are studied. The class of strongly prime graded modules that determines a graded strongly prime radical is described.     

An avoidance principle with an application to the asymptotic behaviour of graded local cohomology

We present an avoidance principle for certain graded rings. As an application we fill a gap in the proof of a result of Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert–Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.     

Prime radicals of graded Ω-groups

In this paper, we introduce the class of graded Ω-groups, which includes: groups; associative, conformal and vertex algebras; Lie algebras and graded algebras. The graded prime radical of a graded Ω-group is defined, and its elementwise characterization is given. It is shown that the graded prime radical of a graded Ω-groups with a finiteness condition coincides with the lower weakly solvable (in the Parfyonov sense) radical. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 159–174, 2006.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

The Castelnuovo regularity of the Rees algebra and the associated graded ring

It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of -sequences.

     

The spherical spectrum of a graded ring

We introduce a functor Sph, the spherical spectrum, which assigns to a graded ringG a space Sph(G) of homogeneous orderings ofG. It combines ideas of concrete geometry in theN-sphere defined by positively homogeneous polynomial equations and inequalities with the abstract notion of the real spectrum of a ring to give a counterpart for real semialgebraic geometry of the functor Proj.     

Graded versions of Goldie’s theorem

We prove graded variants of Goldie’s theorem of existence, structure, and coincidence of right classical quotient ring and right maximal quotient ring of a semiprime (prime) right Goldie’s ring (Theorems 10, 11, and 13). The main difficulty consisting in the problem of existence of a homogeneous regular element in each gr-essential right ideal is solved by posing some additional requirements onto the group grading the ring or onto the homogeneous components of the ring.     

On perfect graded rings

We prove that a group graded ring with finite support is right perfect if and only if the identity graded component is right perfect.     

Monomial modules and graded Betti numbers

Let K be a field, S = K[x ₁,…, x _n ], the polynomial ring over K, and let F be a finitely generated graded free S-module with homogeneous basis. Certain invariants, such as the Castelnuovo-Mumford regularity and the graded Betti numbers of submodules of F, are studied; in particular, the componentwise linear submodules of F are characterized in terms of their graded Betti numbers.     

Basic elements in graded rings

In dealing with graded rings or projective varieties, it is often necessary to find homogeneous elements of graded modules with certain desirable properties. In this paper we prove a “graded version” of the theorem of Eisenbud and Evans on basic elements. [2, Theorem A]. This result is used to generalize a theorem of Kleiman on subbundles of vector bundles on quasiprojective schemes.     

Betti numbers and lifting of Gorenstein codimension three ideals

In this paper we consider homogeneous Gorenstein ideals of codimension three in a polynomial ring and determine their graded Betti numbers in terms of their Hilbert function. For such ideals we prove also a lifting theorem in the vein of a classical result of Hartshorne concerning monomial ideals.     

A radical for graded Lie algebras

For an arbitrary group?G and a G-graded Lie algebra L over a field of characteristic zero we show that the Kostrikin radical of?L is graded and coincides with the graded Kostrikin radical of?L. As an important tool for our proof we show that the graded Kostrikin radical is the intersection of all graded-strongly prime ideals of?L. In particular, graded-nondegenerate Lie algebras are subdirect products of graded-strongly prime Lie algebras.     

A strongly graded ring that is

We construct a ring that is strongly graded by the integers such that it is not graded equivalent to a skew group ring. This is in contrast to the finite case and the results of Cohen and Montgomery, in which every strongly graded ring is graded equivalent to a skew group ring ^{2 2}1991 mathematics subject classification: 16D90, 16S35, 16S40, 16S50, 16W50