Narrow Results

Over a commutative ring the prime spectrum, as a geometrical object, reflects many properties of the ring. This geometrical object can be modified by the action of multiplicative closed subsets, or more in general, by hereditary torsion theories. For a multiplicatively closed subset $S$ of a commutative ring $A$, in the literature there were several $S$-noetherian spectrum properties. In this paper, for any commutative ring $A$, we introduce generalizations of them using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq A$. Two different constructions of noetherian spectra on a ring, associated to a hereditary torsion theory, are considered in such a way that they are characterized by prime ideals and preserved by polynomial ring constructions.

Looking at the automata defined over a group alphabet as a nearring, we see that they are a highly complicated structure. As with ring theory, one method to deal with complexity is to look at semisimplicity modulo radical structures. We find some bounds on the Jacobson 2-radical and show that in certain groups, this radical can be explicitly found and the semisimple image determined.

Each Chinese character is comprised of radicals, where a single character (compound character) contains one (or more than one) radicals. For human cognitive perspective, a Chinese character can be recognized by identifying its radicals and their spatial relationship. This human cognitive law may be followed in computer recognition. However, extracting Chinese character radicals automatically by computer is still an unsolved problem. In this paper, we propose using an improved sparse matrix factorization which integrates affine transformation, namely affine sparse matrix factorization (ASMF), for automatically extracting radicals from Chinese characters. Here the affine transformation is vitally important because it can address the poor-alignment problem of characters that may be caused by internal diversity of radicals and image segmentation. Consequently we develop a radical-based Chinese character recognition model. Because the number of radicals is much less than the number of Chinese characters, the radical-based recognition performs a far smaller category classification than the whole character-based recognition, resulting in a more robust recognition system. The experiments on standard Chinese character datasets show that the proposed method gets higher recognition rates than related Chinese character recognition methods.

Recognizing handwritten Chinese characters is a complex problem. We break the problem down into a series of sub-problems and concentrate on the radical recognition problem. The sub-problems are linked in a hierarchy of three layers: radical, radical sub-unit, and salient stroke. This problem reduction technique enables us to solve complex recognition problems effectively. We describe how to analyze and choose radical sub-units and salient strokes in order to recognize a set of radicals. We also construct two knowledge bases in the form of decision trees to guide the hypothesis and test procedures used to solve the recognition sub-problems. The problem reduction is done in a top-down fashion, while the problem solving process proceeds in a bottom-up fashion. The advantages of our method are described. The representation and organization of the two knowledge bases are explicitly described. The method is tested on two sets of handprinted characters using an IBM PC (486-33). The recognition rate is over 95.7% and the computer time needed to recognize a radical averages about 0.07 s. The experimental results indicate that our method effectively copes with a wide range of ordinary variations encountered in handwritten Chinese characters.

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.

In the paper we pose and discuss new Burnside-type problems, where the role of nilpotency is replaced by that of solvability.

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general.

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜⁿ(Z, X)\ℜⁿ⁺¹(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f₁, f₂)^t : X → Y₁ ⊕ Y₂ is an irreducible epimorphism of finite left degree then we show that d_l(f) = d_l(f₁) + d_l(f₂). We also characterize the artin algebras of finite representation type with α(Γ_A) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.

Let $A={\mathbb{Z}}_p{S}_n$ with $p$ a prime and $S_n$ a symmetric group. We prove in this paper that if $n<2p$, then $\mathrm{\text{rad}}A=I$, where $I$ is the nilpotent ideal constructed in [Radicals of symmetric cellular algebras, Collog. Math.133 (2013) 67–83]. Finally we give two remarks on algebras $A={\mathbb{Z}}_p{S}_n$ with $n<2p$.

According to Albu and Iosif, [2, Definition 1.1] a lattice preradical is a subfunctor of the identity functor on the category ${{ℒ}}_{{ℳ}}$ of linear modular lattices, whose objects are the complete modular lattices and whose morphisms are linear morphisms. In this paper, we describe some big lattices which are isomorphic to the big lattice of lattice preradicals and we study the four classical operations that occur in the lattice of preradicals of modules over a ring $R$, namely, the join, the meet, the product and the coproduct. We show that some results about the lattice of module preradicals can be extended to the lattice of lattice preradicals. In particular, we show the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical $\sigma$, as well as some of their properties.

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical.

We report a theoretical study of the cyclopropanation reactions of EtZnCHI, (EtZn)₂CH EtZnCHZnI, and EtZnCIZnI radicals with ethylene. The mono-zinc and gem-dizinc radical carbenoids can undergo cyclopropanation reactions with ethylene via a two-step reaction mechanism similar to that previously reported for the CH₂I and IZnCH₂ radicals. The barrier for the second reaction step (ring closure) was found to be highly dependent on the leaving group of the cyclopropanation reaction. In some cases, the (di)zinc carbenoid radical undergoes cyclopropanation via a low barrier of about 5–7 kcal/mol on the second reaction step and this is lower than the CH₂I radical reaction which has a barrier of about 13.5 kcal/mol for the second reaction step. Our results suggest that in some cases, zinc radical carbenoid species have cyclopropanation reaction barriers that can be competitive with their related molecular Simmons-Smith carbenoid species reactions and produce somewhat different cyclopropanated products and leaving groups.

The complex singlet and triplet potential energy surfaces (PESs) of the [C₂N₂O₂] system are performed at the B3LYP and Gaussian-3//B3LYP levels in order to investigate the possibility of the carbyne radical CCN in removal of nitrogen dioxide. Thirty minimum isomers and 36 transition states are located. Starting from the very energy-rich reactant RCCN + NO₂, the terminal C-attack adduct NCCN(O)O (singlet at -48.6 and triplet at -48.1 kcal/mol) is first formed on both singlet and triplet PESs. Subsequently, the singlet NCCN(O)O takes an O-transfer to form the intermediate singlet cis-NCC(O)NO (-120.1), which can lead to the fragments NCCO + NO (-94.4) without barrier. The simpler evolution of the triplet NCCN(O)O is the direct N–O rupture to form the final fragmentation NCCNO + ³O (-31.0). However, the lower lying products ³NCNO + CO (-103.3) and NCNCO + ³O (-86.5) are kinetically much less competitive. All the involved transition states for the generation of NCCO + NO and NCCNO + ³O lie much lower than the reactants, and it indicates that this reaction can proceed effectively even at low temperatures. We expect that the reaction CCN + NO₂ can play a role in both combustion and interstellar processes. Comparison is made between the CCN + NO₂ and CH + NO₂ reaction mechanisms.

The reaction of N (⁴S) radical with NCO (X²Π) radical has been studied theoretically using density functional theory and ab initio quantum chemistry method. The triplet electronic state [N₂CO] potential energy surface (PES) is calculated at the G3B3 and CCSD(T)/aug-cc-pVDZ//B3LYP/6-311++G(d,p) levels of theory. All the energies of the transition states and isomers in the pathway RP1 are lower than that of the reactants; the rate of this pathway should be very fast. Thus, the novel reaction N + NCO can proceed effectively even at low temperatures and it is expected to play a role in both combustion and interstellar processes. On the basis of the analysis of the kinetics of all pathways through which the reactions proceed, we expect that the competitive power of reaction pathways may vary with experimental conditions for the title reaction.

The structures, vibrational frequencies and adiabatic ionization energies of HPCN and HNCP are calculated at several levels of theory. The adiabatic ionization energy of HPCN is found to be 10.16 eV at the G3 level of theory. The singlet state of HPCN⁺ is found to be approximately 1.3 eV below the lowest energy triplet state (ã³A″). Both states have a bent equilibrium molecular geometry. The adiabatic ionization energy for HNCP is calculated to be 8.30 eV at the G3 level of theory. In contrast to HPCN⁺, the triplet state of HNCP⁺ is lower in energy than that of the singlet state (ã¹A′) by approximately 1 eV. Also, the triplet state of HNCP⁺ is linear in contrast to that of HPCN⁺ due to the larger interaction between neighboring 2p orbitals on the central atoms in HNCP⁺ relative to the interaction between the 3p and 2p orbitals on the central atoms in HPCN⁺. Simulated photoelectron spectra (PES) are presented for the transitions producing both the singlet and triplet ion states of both isomers. As predicted by the dramatic geometry change in the case of , there is a long progression in the bending mode of the cation in the simulated PES.

For near-ring ideal mappings ρ₁ and ρ₂, we investigate radical theoretical properties of and the relationship among the class pairs (ρ₁: ρ₂), and (ℛ_ρ2: ℛ_ρ1). Conditions on ρ₁ and ρ₂ are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that forms a KA-radical class, where denotes the class of completely prime near-rings and the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.