V-Categories: Applications to Graded Rings

Motivated by the study of V-rings, we introduce the concept of V-category, as a Grothendieck category with the property that any simple object is injective. We present basic properties of V-categories, and we study this concept in the special case of locally finitely generated categories, for instance the category R-gr of all graded left R-modules, where R is a graded ring. We use the characterizations of V-categories in the study of graded V-rings. Since V-rings are closely related to Von Neumann regular rings (in the commutative case these classes of rings coincide), the last part of the article is devoted to graded regular rings.     

关于分次本质右理想

设 Ｒ是 Ｇ－分次,本文讨论了环 Ｒ的相关环 Ｒ,Ｒ＃ Ｇ^*, Ｒｅ, Ｑ（Ｒ）, Ｒ^G, Ｒ*Ｇ及 Ｒ的正规化扩张Ｓ的非奇异性,右一致性,右基座之间的关系．当Ｒ_Ｒ是ＹＪ－内射模时,证明了Ｊ（Ｒ）＝Ｚ（Ｒ）。     

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

Injective hulls with distinct ring structures

It is well known from Osofsky’s work that the injective hull E(R_R) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(R_R) has a ring structure and its multiplication extends its R-module scalar multiplication, must the ring structure be unique? In this paper, we utilize the properties of Morita duality to explicitly describe an injective hull of a ring R with R=Q(R) (where Q(R) is the maximal right ring of quotients of R) such that every injective hull of R_R has (possibly infinitely many) distinct compatible ring structures which are mutually ring isomorphic and quasi-Frobenius. Further, these rings have the property that the ring structures for E(R_R) also are ring structures on E(_RR).     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.     

On the depth of certain graded rings associated to an ideal

Let (R,m) be a local ring and Ian ideal of R. In this work we find conditions on Ithat allow us to describe simple relations among depth R(It), depth gr_I(R), depth S(I) and depth S(I/I ²). These relations are useful also from a practical point, of view since it is usually difficult to evaluate depth gr_I(R) and depth S(I/I ²) even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class     

Continuous symmetrized Sobolev inner products of order N (I)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies that Q2n(x)=Pn(x²), Q2n+1(x)=xRn(x²) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper, we deduce the integral representation of the inner products such that {Pn} and {Rn} are the corresponding sequences of orthogonal polynomials. Moreover, we state a relation between both inner products which extends the classical result for symmetric linear functionals.     

On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions

We study the differential equations w ²+R(z)(w ^(k))² = Q(z), where R(z),Q(z) are nonzero rational functions. We prove

1. if the differential equation w ²+R(z)(w′)² = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
2. if the differential equation w ² + R(z)(w ^(k))² = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .

     

Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

The Graded Cobordism Group of Codimension-One Immersions

The cobordism group N(Mⁿ) of codimension-one immersions in the n-manifold Mⁿ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group gr*N(Mⁿ). We introduce some new invariants for immersions enlightening the Atiyah–Hirzebruch spectral sequence associated to N(M), which are of combinatorial-geometric nature. Explicit computations are developed for n 7, and the group structure is also investigated for orientable 4-manifolds.     

Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this article, we give new characterizations of the Buchsbaum and Cohen–Macaulay properties of the tangent cone gr _𝔪 (R), where (R, 𝔪) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr _𝔪 (R).     

Oscillation results for selfadjoint differential systems

We consider the second-order differential system, (1) (R(t) Y′)′ + Q(t) Y = 0, where R, Q, Y are n × n matrices with R(t), Q(t) symmetric and R(t) positive definite for t ϵ [a, + ∞) (R(t) > 0, t ⩾ a). We establish sufficient conditions in order that all prepared solutions Y(t) of (1) are oscillatory; that is, det Y(t) vanishes infinitely often on [a, + ∞). The conditions involve the smallest and largest eigen-values λ_n(R⁻¹(t)) and λ₁(∝_a^t Q(s) ds), respectively. The results obtained can be regarded as generalizing well-known results of Leighton and others in the scalar case.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.     

Cocharacters of Bilinear Mappings and Graded Matrices

Let M _k (F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow M _k (F) with the elementary grading induced by the k-tuple (1,...,1,g) where g?∈?G, g ²?≠?1. Then the graded identities of M _k (F) depending only on variables of homogeneous degree g and g ^???1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g ^???1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles.     

Multivariable monotonic optimization over multivalued logics and rectangular design lattices

A minimum dichotomous direct search procedure is given for finding the optimum combination of N variables, each having M(n) possible values, when a certain monotonicity condition is satisfied. The least upper bound on the number of objective function evaluations is 1 + Σ^N_n=1Q(n), where Q(n) is defined by 2^Q(n)-1<M(n) < 2^Q(n), whereas the total number of possibilities is Π^N_R=1M(n). An example shows where the procedure applies to restricted problems in multivalued logic and engineering design.     

On Lefschetz series

Let R = ⊕ _i=0^∞ R ⁱ be a connected graded commutative algebra over the field ℚ of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f) on the ℚ vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f). The main tool we used is the graded algebra version of Gr?bner basis theory. At the end of this paper, some examples and applications are given.     

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension O of a two-dimensional local regular ring (R,m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X=BlJ(R) has some point Q whose local ring OX,Q is analytically isomorphic to O.     

A simplification of the Kovarik formula

Let P,Q be two idempotents on a Hilbert space. Z.V. Kovarik (Z.V. Kovarik, Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977) 341-351) showed that when P+Q−I is invertible, the formula K(P,Q)=P⁻²(P+Q−I)Q gives the only idempotent such that R(K)=R(P), N(K)=N(Q), where N(T) and R(T) denote the nullspace and the range of a bounded linear operator T on a Hilbert space, respectively. This formula was later extended to the context of Banach algebras and used in 1983 by J. Esterle to show that two homotopic idempotents may always be connected by a polynomial idempotent valued path. In the present paper, we give a simplification of Kovarik's original formula and one natural generalization of it.     

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, SV, and the multi-index graded algebra defined by V, grVR. We prove that SV is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V(k) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V(k), allow us to obtain the (finite) minimal generating sequences for C as well as for V.We also analyze the structure of the homogeneous components of grVR. The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V(k), k?0.