Theorem For Rings Research Articles

Semifinite Harmonic Functions on the Gnedin–Kingman Graph

We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem.

On the Briançon-Skoda theorem for analytic local rings with singularities

In this note, we study validity of the original Briançon-Skoda theorem for analytic local rings. By introducing a new concept of multiplier ideal sheaves associated to plurisubharmonic functions on any (reduced) complex space of pure dimension, we establish the original Briançon-Skoda theorem for normal weakly rational singularities (not essentially of finite type over C and Cohen-Macaulay). Moreover, we also characterize two dimensional rational singularities by the original Briançon-Skoda theorem.

Second cohomology of multiplicative Lie rings and Schreier's extension theory

Our goal of this paper is to develop an analogue of the theory of group extensions for multiplicative Lie rings. We first define a factor system of pair of multiplicative Lie rings , which use to construct an extension of by . Then we state Schreier's theorem for multiplicative Lie rings. We also use this notation to introduce second cohomology group of multiplicative Lie rings. Finally, we show that the equivalence classes of multiplicative Lie ring extensions can be identified with second cohomology group , where acts on by and .

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

Higher derivations and Posner’s second theorem for semiprime rings

Let $$(d_i)_{i\in \mathbb {N}}$$ and $$(\delta _j)_{j\in \mathbb {N}}$$ be two higher derivations of order m and n respectively on semiprime ring R such that $$d_m(x)x-x\delta _n(x)\in Z(R)$$ for all $$x\in I$$ , where I be an ideal of R. In this article we generalize Posner’s second theorem for higher derivation and prove that either R is commutative or some linear combination of $$(\delta _j)$$ sends Z(R) to zero.

Lattice Burnside rings

Given a finite group G and a finite G-lattice $${{\mathscr {L}}}$$ , we introduce the concept of lattice Burnside ring associated to a family of nonempty sublattices $${{\mathscr {L}}}_H$$ of $${{\mathscr {L}}}$$ for $$H\le G$$ . The slice Burnside ring introduced by Bouc is isomorphic to a lattice Burnside ring. Any lattice Burnside ring is an extension of the ordinary Burnside ring and is isomorphic to an abstract Burnside ring. The ring structure of a lattice Burnside ring is explored on the basis of the fundamental theorem for abstract Burnside rings. We explore the unit group, the primitive idempotents, and connected components of the prime spectrum of a lattice Burnside ring. There are certain abstract Burnside rings called partial lattice Burnside rings. Any partial lattice Burnside ring consists of elements of a lattice Burnside ring. The section Burnside ring introduced by Bouc, which is a subring of the slice Burnside ring, is isomorphic to a partial lattice Burnside ring.

When proper cyclics are homomorphic image of injectives

Quasi-Frobenius rings are precisely rings over which any right module is a homomorphic image of an injective module. We investigate the structure of rings whose proper cyclic right modules are homomorphic image of injectives. The class of such rings properly contains that of right self-injective rings. We obtain some structure theorems for rings satisfying the said property and apply them to the Artin algebra case: It follows that an Artin algebra with this property is Quasi-Frobenius.

A Cohen-type theorem for w-Artinian modules

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

Uniquely separable extensions

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1 (centrally projective extension). Separable and split extensions have separability idempotents and bimodule projections in 1–1 correspondence via an endomorphism ring theorem in Section 3. If the separable idempotent is unique, then the separable extension is called uniquely separable. A Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier group-theoretic results are recovered and related to depth 1. The dual notion, uniquely split extension, only occurs trivially for finite group algebra extensions over complex numbers.

On the Discretized Sum-Product Problem

Abstract We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset \mathbb {R}$ is a $(\delta ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac {1}{68}}$.

Virtually uniserial modules and rings

We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0≠K⊆M, K/Rad(K) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bézout. Also, we show that if R is a left virtually serial ring, then R/J(R)≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J(R)=0 if and only if R≅∏i=1tMni(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain with J(Di)=0. Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial.

Eversible and reversible semigroups and semirings

The main purpose of this paper is to investigate the zero-divisors of semigroups with zero and semirings. In particular, we discuss eversible and reversible semigroups and semirings. We also introduce a new ring-like algebraic structure called prenearsemiring and generalize Cohn’s theorem for reversible rings in this context. Finally, we discuss when expectation semirings are reversible or eversible.

Detection and Localization of Submodule Open-Circuit Failures for Modular Multilevel Converters With Single Ring Theorem

Submodule (SM) failure detection and localization is crucial to reliability improvement of modular multilevel converters (MMCs), which consist of numerous identical SMs. Upon SM failures, the capacitor voltages of the faulty SMs deviate from those of the healthy SMs. Hence, SM voltage consistency disruption can be utilized as an indicator of SM failures. To enable effective consistency evaluation of numerous SM voltages, the single ring theorem, which is an ideal analyzing tool for large size matrices, is applied to SM failure detection in this paper. The proposed SM failure detection method eliminates the need for ideal-state estimators and thus ensures sufficient robustness in terms of parameter uncertainties. Additional voltage/current sensors are not required, which is beneficial for cost efficiency. A statistical-analysis-based SM failure localization method enabling fast identification of the faulty SMs upon failure detection is also proposed. Both single and multiple SM failure detection and localization can be handled with the proposed method even if the faulty SMs are in different arms. The effectiveness of the proposed SM failure diagnosis method is verified by both simulation in MATLAB/Simulink and experimentation on a 13-level MMC prototype.

Some commutativity theorems for rings with involution involving generalized derivations

Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

On commutative rings whose ideals are direct sums of cyclic modules

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension [Formula: see text]. Then we conclude that, for a ring of local dimension [Formula: see text], every ideal of [Formula: see text] is a direct sum of cyclic modules if and only if every indecomposable ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if [Formula: see text] is a principal ideal ring.

Multiplicative de Rham Theorems for Relative and Intersection Space Cohomology

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space (co-)homology is a modified (co-)homology theory extending Poincar\'e Duality to stratified pseudomanifolds. The novelty of our result compared to the de Rham isomorphism given previously by Banagl is, that we indeed have an isomorphism of rings and not just of graded vector spaces. We also provide a proof of the de Rham Theorem for cohomology rings of pairs of smooth manifolds which we use in the proof of our main result.

Local single ring theorem on optimal scale

Let $U$ and $V$ be two independent $N$ by $N$ random matrices that are distributed according to Haar measure on $U(N)$. Let $\Sigma$ be a nonnegative deterministic $N$ by $N$ matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix $X:=U\Sigma V^{*}$ converges weakly, in the limit of large $N$, to a deterministic measure which is supported on a single ring centered at the origin in $\mathbb{C}$. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order $N^{-1/2+\varepsilon}$ and establish the optimal convergence rate. The same results hold true when $U$ and $V$ are Haar distributed on $O(N)$.

English

Let $$\mathfrak{a}$$ , I, J be ideals of a Noetherian local ring $$(R,\mathfrak{m},k)$$ . Let M and N be finitely generated R-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $$H^t_{I,J}$$ (M) and $$H^t_{I,J}$$ (M)), where t is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and D(−):= HomR(−, Er(k)) is the Matlis dual functor. We show that if R is a d-dimensional complete Cohen-Macaulay ring and $$H^i_{I,J}$$ (R) = 0 for all i ≠ t, the natural homomorphism R → Homr( $$H^t_{I,J}$$ (KR), $$H^t_{I,J}$$ (KR)) is an isomorphism, where KR denotes the canonical module of R. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.

MacWilliams’ extension theorem for infinite rings

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

Compatibility of Quasi-Orderings and Valuations: A Baer-Krull Theorem for Quasi-Ordered Rings

In his work of 1969, Merle E. Manis introduced valuations on commutative rings. Recently, the class of totally quasi-ordered rings was developped by the second author. In the present paper, we establish the notion of compatibility between valuations and quasi-orders on rings, leading to a definition of the rank of a quasi-ordered ring. Moreover, we prove a Baer-Krull Theorem for quasi-ordered rings: fixing a Manis valuation v on R, we characterize all v-compatible quasi-orders of R by lifting the quasi-orders from the residue class ring to R itself.