Trivial Ring Research Articles

Topological constraint in ring polymers under theta conditions studied by Monte Carlo simulation

We studied equilibrium conformations of trivial-, 3(1)-, and 5(1)-knotted ring polymers together with a linear counterpart over the wide range of segment numbers, N, from 32 up through 2048 using a Monte Carlo simulation to obtain the dependence of the radius of gyration of these simulated polymer chains, R(g), on the number of segments, N. The polymer chains treated in this study are composed of beads and bonds placed on a face-centered-cubic lattice respecting the excluded volume. The Flory's critical exponent, ν, for a linear polymer is 1/2 at the θ-temperature, where the excluded volume is screened by the attractive force generated among polymer segments. The trajectories of linear polymers at the θ-condition were confirmed to be described by the Gaussian chain, while the ν values for trivial-, 3(1)-, and 5(1)-knots at the θ-temperature of a linear polymer are larger than that for a linear chain. This ν value increase is due to the constraint of preserving ring topology because the polymer chains dealt with in this study cannot cross themselves even though they are at the θ-condition. The expansion parameter, β, where N-dependence vanishes by the definition, for trivial-, 3(1)-, and 5(1)-knotted ring polymers is obtained at the condition of ν = 1/2. It has been found that β decreases with increasing the degree of the topological constraint in the order of trivial (0.526), 3(1) (0.422), and 5(1) knot (0.354). Since the reference β value for a random knot is 0.5, the trivial ring polymer is swollen at ν = 1/2 and the other knotted ring polymers are squeezed.

On P-semihereditary rings

In this paper, we introduce the notion of “P-semihereditary rings” which is a generalization of the notion of semihereditary rings. We establish the transfer of this notion to trivial ring extensions and provide a class of P-semihereditary rings which are not a semihereditary rings.

Left Noetherian rings with differentially trivial proper quotient rings

Left Noetherian rings with differentially trivial proper quotient rings

Un anneau de déformation universel en conducteur supérieur

Let k denote a perfect field of characteristic 5. We show that the versal deformation ring of an element of order 5 and Hasse conductor 2 as automorphism of a ring of formal power series k½½ t�� , computed by Bertin and Mezard, is in fact universal. This provides the first example of a non trivial universal deformation ring in higher conductor.

Synthesis, separation and characterization of knotted ring polymers

Knotted ring polystyrene (PS) with molecular weight of 380 k was successfully synthesized by intramolecular cyclization reaction in cyclohexane under extremely diluted condition. Crude product was confirmed to include linear precursor molecule, single ring molecules, and various intermolecular-reacted byproducts by SEC and interaction chromatography characterizations. The crude product was fractionated repeatedly several times by high performance interaction chromatography and finally highly-purified knotted ring molecules were obtained. It has been found by SEC-MALS that the chain dimension of the knotted ring polymers is evidently smaller than those of linear and the trivial ring polymer, while knotted polymer molecules have the same absolute molecular weight as the corresponding counterparts.

On FF-rings

In this paper, we study the class of rings in which every flat ideal is finitely generated. We investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as direct products, pullback rings, and trivial ring extensions. Our results generate examples which enrich the current literature with new and original families of non-Noetherian rings that satisfy this property.

An open-image theorem for a general class of abelian varieties

Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension 1, then, for almost all rational primes ℓ, the Galois group of the splitting field of the ℓ-torsion of A is GSp2g(ℤ/ℓ).

Commutative rings in which every finitely generated ideal is quasi-projective

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni–Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni–Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.

On Strong (A)-Rings

In this paper, we introduce a strong property (A) as follows: A ring R is called satisfying strong property (A) if every finitely generated ideal of R which is generated by a finite number of zero-divisors elements of R, has a non zero annihilator. We study the transfer of property (A) and strong property (A) in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings which satisfy property (A) and do not satisfy strong property (A).

Tilting and trivial extensions

The trivial extensions of a quasi-abelian category by means of a fully exact endofunctor are again quasi-abelian. Using the one-to-one correspondence between quasi-abelian categories and tiltings, it is shown that the trivial extension of a quasi-abelian category corresponds to a trivial extension of the associated tilting. As an application, a criterion for tilting modules over an arbitrary ring R to be liftable to a tilting module over a trivial extension ring \({R\ltimes M}\) is given.

Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.

Trivial extensions defined by Prüfer conditions

This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni–Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky–Tsang conjecture on the content ideal of Gaussian polynomials.

Trivial source bimodule rings for blocks and $p$-permutation equivalences

We associate with any p-block of a finite group a Grothendieck ring of certain p-permutation bimodules. We extend the notion of p-permutation equivalences introduced by Boltje and Xu [4] to source algebras of p-blocks of finite groups. We show that a p-permutation equivalence between two source algebras A, B of blocks with a common defect group and same local structure induces an isotypy.

A generalization of von Neumann regularity

We propose two theories, one generalizing the notion of regularity, the other symmetric to it. Under two additional axioms (as for the Carson–Lipshitz–Saracino theorems) one obtains model completeness of both theories. Models of these theories can be viewed as rings of sections (over a boolean space) of sheaves whose stalks are valuation rings. Regular rings correspond to the special case where all stalks are trivial valuation rings, that is fields.

On Weakly Finite Conductor Rings#

We define a weakly finite conductor rings with zero divisors, and examine the transfer of these rings to the trivial ring extensions. These results provide examples of a weakly finite conductor rings that are not finite conductor rings (and so not coherent rings). The last section show that the class of weakly finite conductor rings and the class of 2-coherent rings are not comparable.

The effective surjectivity of mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring

Mod l Galois representations of 1- and 2-dimensional abelian varieties with trivial endomorphism ring are surjective for sufficiently large prime l as Serre proved. But he did not give an effective lower bound of l 0 such that they are surjective for l > l 0. We supply an effective evaluation of l 0 by an ”elementary” proof of the surjectivity. The proof uses the Masser-Wustholz theorem and Kleidman and Liebeck’s classification of the maximal subgroups of GL 2 F l ) and GSp 4 (F l ).

A Note on Some (3, 2K + 1)-Associative Rings

In this paper we consider the associativity of a (3, 2k + 1)-associative ring R in the following cases: (1) R is simple 2-divisible; (2) R is p-divisible trivial right ideal ring; (3) R is prime p-divisible.

Lifting Galois representations and a conjecture of Fontaine and Mazur

Mumford has constructed 4-dimensional abelian varieties with trivial endomorphism ring, but whose Mumford–Tate group is much smaller than the full symplectic group. We consider such an abelian variety, defined over a number field F , and study the associated p -adic Galois representation. For F sufficiently large, this representation can be lifted to \mathbf{G}_m(\mathbf{Q}_p)\times\mathrm{SL}_2(\mathbf{Q}_p)^3 . Such liftings can be used to construct Galois representations which are geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question predicts that these representations should come from algebraic geometry. We confirm the conjecture for the representations constructed here.

Differentially Trivial and Rigid Rings of Finite Rank

In this paper we study the rings of finite rank which have only trivial derivations (respectively endomorphisms). The main results are characterizations of these rings.

Topologically Driven Compatibility Enhancement in the Mixtures of Rings and Linear Chains

The thermodynamics of the bicomponent melts of linear chains A and unknotted unentangled closed chains B (trivial rings) is considered within the framework of the usual Flory approach. It is shown that such melts exhibit a compatibility enhancement in comparison with the corresponding melts of linear chains. The reason for such effect is due to the topological entropy gain of mixing of A- and B-chains : if B-chains are unentangled since the A-chains play the role of a diluent softening the topological restrictions imposed on the B-chains. This result is in agreement with recent experimental findings of McKnight et al.