Product Of Rings Research Articles

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$-modules. Assume $n$ is a nonnegative integer and that the tensor product $M\tensor_{R}N$ is an $(n+c)$th syzygy of some finitely generated $R$-module. If Tor$^{R}_{>0}(M,N)=0$, then both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules.

Some Maximal Commutative Subrings of M n (D)

We answer a question of Weibel and show that maximal commutative subgrings of Artinian semisimple rings are direct products of local rings.

Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles

Given an n-tuple of positive real numbers α we consider the hyperpolygon space X(α), the hyperkähler quotient analogue to the Kähler moduli space of polygons in R3. We prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-2, holomorphically trivial parabolic Higgs bundles over CP1 with fixed determinant and trace-free Higgs field. This isomorphism allows us to prove that hyperpolygon spaces X(α) undergo an elementary transformation in the sense of Mukai as α crosses a wall in the space of its admissible values. We describe the changes in the core of X(α) as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles. Moreover, we study the intersection rings of the core components of X(α). In particular, we find generators of these rings, prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers. Using our isomorphism, we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles thus determining their intersection rings. As a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic Higgs bundles and of the components of their nilpotent cone.

Commutative Local Rings Whose Ideals are Direct Sums of Cyclic Modules

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257–265, 2011) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.

ON GROMOV–WITTEN INVARIANTS OF DEL PEZZO SURFACES

We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81–148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on [Formula: see text], the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type [Formula: see text], the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of [Formula: see text] via enumerative invariants of [Formula: see text], using Vakil's extension of the Abramovich–Bertram formula.

Prime Graph of Cartesian Product of Rings

Let be a commutative ring. The prime graph of the ring is defined as a graph whose vertex set consists of all elements of and any two distinct vertices x and y are adjacent if and only if or . This graph is denoted by . In this paper we investigate some relations between the chromatic number of prime graph of finite product of commutative rings and the chromatic number of prime graph of these rings. We also obtain some results on the chromatic number of prime graph of the ring .

Comultiplication Modules over Noncommutative Rings

Comultiplication Modules over Noncommutative Rings

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

The main purpose of this paper is to study the zero-divisor graph for direct product of finite commutative rings. In our present investigation we discuss the zero-divisor graphs for the following direct products: direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p for a prime number p, direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo 2p for an odd prime number p and direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p – 2 for that odd prime p for which p – 2 is a prime number. The aim of this paper is to give some new ideas about the neighborhood, the neighborhood number and the adjacency matrix corresponding to zero-divisor graphs for the above mentioned direct products. Finally, we prove some results of annihilators on zerodivisor graph for direct product of A and B for any two commutative rings A and B with unity

Schur rings over a product of Galois rings

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring \(R\) being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of \(R\) is odd, then as in the cyclic group case any pure Schur ring over \(R\) is the tensor product of a pure cyclotomic ring and Schur rings of rank \(2\) over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over \(R\) that are not generalized wreath products.

On Dixmier–Duflo isomorphism in positive characteristic – The classical nilpotent case

Let g be the nil radical of the Borel subalgebra of one of the classical simple Lie algebras over a field F of characteristic p⩾0. For p>0 we find an explicit realization of the center Z(g) of the enveloping algebra U(g) by generators and relations. This constructive approach yields an explicit isomorphism between Z(g) and the polynomial invariants algebra S(g)g. While realizing Z(g), we also prove that Z(g) is a complete intersection ring. Moreover, it leads to an explicit realization of Z(g) and S(g)g for p=0 as well. This extends a result of Dixmier in type An.

Novel cyclotriphosphazene-based epoxy compound and its application in halogen-free epoxy thermosetting systems: Synthesis, curing behaviors, and flame retardancy

Abstract A novel cyclotriphosphazene-based epoxy compound (PN-EPC) as a halogen-free reactive-type flame retardant was synthesized via a two-step synthetic route. The chemical structures and compositions of the cyclotriphosphazene precursor and the final product were characterized by 1 H, 13 C, and 31 P NMR spectroscopy, Fourier transform infrared spectroscopy, mass spectroscopy, and elemental analysis. A series of thermosetting systems based on a conventional epoxy resin and PN-EPC were prepared, and their thermal curing behaviors were investigated. These epoxy thermosets achieved a significant improvement in glass transition temperature and also gained the good thermal stability with a high char yield. The incorporation of PN-EPC could impart an excellent nonflammability to the epoxy thermosets due to a synergistic flame retarding effect as a result of the unique combination of phosphorus and nitrogen from the phosphazene rings, and these epoxy thermosets achieved the high limiting oxygen indexes and the UL-94 V-0 rating when 20 wt.% of PN-EPC was added. The study on flame-retardant mechanism indicates that the pyrolysis products of phosphazene rings acted in both the condensed and gaseous phases to promote the formation of intumescent phosphorus-rich char on the surface of the epoxy thermosets. Such a char layer can supply a much better barrier for underlying thermosets to inhibit gaseous products from diffusing to the flame, to shield the surface of the thermosets from heat and air, and to prevent or slow down oxygen diffusion. As a result, the resulting epoxy thermosets obtained an excellent nonflammability.

An inductive approach to representations of complex reflection groups G(m, 1, n)

An inductive approach to the representation theory of the chain of the complex reflection groups G(m,1,n) is presented. We obtain the Jucys-Murphy elements of G(m,1,n) from the Jucys--Murphy elements of the cyclotomic Hecke algebra, and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. Representations of G(m,1,n) are constructed with the help of a new associative algebra whose underlying vector space is the tensor product of the group ring of G(m,1,n) with a free associative algebra generated by the standard m-tableaux.

SEMICOPRIME PRERADICALS

We define the concept of "semicoprime" for preradicals and for submodules and we prove some properties that relate both of them. For any ring we define the ultrasocle preradical as a certain join of maximal semicoprime preradicals. It defines a kind of primary decomposition on modules. We compare the greatest semicoprime preradical, the meet of all unipotent preradicals, the socle preradical, and the ultrasocle preradical. We characterize rings which are finite product of simple rings in terms of some of these preradicals. We study the least semicoprime preradical above any preradical and we prove some of its properties. Using "Amitsur constructions" we define some related operators and prove some of their properties.

On Maximal Subrings of Commutative Rings

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then | Max (R)| ≤ | RgMax (R)|, where RgMax (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

Construction and decoding of BCH codes over chain of commutative rings

Abstract In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A 0 ⊂ A 1 ⊂ ⋯ ⊂ A t − 1 ⊂ A t be a chain of unitary commutative rings, where each A i is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K 0 ⊂ K 1 ⊂ ⋯ ⊂ K t − 1 ⊂ K t (another chain of unitary commutative rings), where each K i is made by the direct product of corresponding residue fields of given Galois rings. Also, A i ∗ and K i ∗ are the groups of units of A i and K i , respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A i ∗ and K i ∗ for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.

Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD’s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⟺ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx 1 = x then there is a y 1 ∈ S with dy 1 = y and x 1 ∧ y 1 = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.

A ghost algebra of the double Burnside algebra in characteristic zero

For a finite group G, we introduce a multiplication on the Q-vector space with basis SG×G, the set of subgroups of G×G. The resulting Q-algebra à can be considered as a ghost algebra for the double Burnside ring B(G,G) in the sense that the mark homomorphism from B(G,G) to à is a ring homomorphism. Our approach interprets QB(G,G) as an algebra eAe, where A is a twisted monoid algebra and e is an idempotent in A. The monoid underlying the algebra A is again equal to SG×G with multiplication given by composition of relations (when a subgroup of G×G is interpreted as a relation between G and G). The algebras A and à are isomorphic via Möbius inversion in the poset SG×G. As an application we improve results by Bouc on the parametrization of simple modules of QB(G,G) and also of simple biset functors, by using results of Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where G is a cyclic group of order n, we give an explicit isomorphism between QB(G,G) and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order k, where k divides n.

Fundamental Reaction Pathway and Free Energy Profile for Inhibition of Proteasome by Epoxomicin

First-principles quantum mechanical/molecular mechanical free energy calculations have been performed to provide the first detailed computational study on the possible mechanisms for reaction of proteasome with a representative peptide inhibitor, Epoxomicin (EPX). The calculated results reveal that the most favorable reaction pathway consists of five steps. The first is a proton transfer process, activating Thr1-O(γ) directly by Thr1-N(z) to form a zwitterionic intermediate. The next step is nucleophilic attack on the carbonyl carbon of EPX by the negatively charged Thr1-O(γ) atom, followed by a proton transfer from Thr1-N(z) to the carbonyl oxygen of EPX (third step). Then, Thr1-N(z) attacks on the carbon of the epoxide group of EPX, accompanied by the epoxide ring-opening (S(N)2 nucleophilic substitution) such that a zwitterionic morpholino ring is formed between residue Thr1 and EPX. Finally, the product of morpholino ring is generated via another proton transfer. Noteworthy, Thr1-O(γ) can be activated directly by Thr1-N(z) to form the zwitterionic intermediate (with a free energy barrier of only 9.9 kcal/mol), and water cannot assist the rate-determining step, which is remarkably different from the previous perception that a water molecule should mediate the activation process. The fourth reaction step has the highest free energy barrier (23.6 kcal/mol) which is reasonably close to the activation free energy (∼21-22 kcal/mol) derived from experimental kinetic data. The obtained novel mechanistic insights should be valuable for not only future rational design of more efficient proteasome inhibitors but also understanding the general reaction mechanism of proteasome with a peptide or protein.

RINGS WHICH ADMIT FAITHFUL TORSION MODULES II

Let R be an associative ring with identity. An (left) R-module M is said to be torsion if for every m ∈ M, there exists a nonzero r ∈ R such that rm = 0, and faithful provided rM = {0} implies r = 0 (r ∈ R). We call R (left) FT if R admits a nontrivial (left) faithful torsion module. In this paper, we continue the study of FT rings initiated in Oman and Schwiebert [Rings which admit faithful torsion modules, to appear in Commun. Algebra]. After presenting several examples, we consider the FT property within several well-studied classes of rings. In particular, we examine direct products of rings, Brown–McCoy semisimple rings, serial rings, and left nonsingular rings. Finally, we close the paper with a list of open problems.

Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD’s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⟺ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx1 = x then there is a y1 ∈ S with dy1 = y and x1 ∧ y1 = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.