Journal of Algebra and Its Applications

We provide a counterexample to Theorem 2.6 in a Ziembovski paper.

We study semiprime superalgebras with superinvolution whose symmetric or skewsymmetric elements are assumed to be regular or nilpotent, and superalgebras in which xx* = 0 implies x*x = 0.

Given an order ideal 𝒪 and an 𝒪-border basis of a 0-dimensional polynomial ideal, it was shown by Huibregtse that the liftings of the neighbor syzygies (i.e. of the fundamental syzygies of neighboring border terms) form a system of generators for the syzygy module of the border basis. We elaborate on Huibregtse's proof and transform it into explicit algorithmic form. Based on this, we are able to exhibit explicit conditions on a module term ordering τ such that the liftings of the neighbor syzygies are in fact a τ-Gröbner basis. Finally, we construct term orderings satisfying these conditions in an explicit algorithmic way.

Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set M_n(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = ℝ and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

Using Buchberger–Shirshov Algorithm, Composition–Diamond Lemma and partitions of integers we obtain the reduced Gröbner–Shirshov basis of Ã_n and classify all reduced words of the affine Weyl group Ã_n.

The finite groups G such that, whenever H < G and |H|² < |G|, then H ◃ G are classified in Theorems 2.1 and 3.2. Let p^ν(X)(p^a(X)) be the maximal order of an abelian (a minimal nonabelian) subgroup of a p-group X. We classify the p-groups G such that, whenever an abelian H < G and |H| ≤ p^ν(G)-2, then H ◃ G. Next, we describe the p-groups G all of whose abelian subgroups of order ≤ p^a(G)-2 are normal. Minimal nonabelian and minimal nonnilpotent groups play crucial role in the whole paper.

In this paper, we will extend the notion of zero-divisor graph of commutative rings to zero-divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero-divisor graph of almost all abelian groups. Just a few classes of reduced abelian groups remain untouched in this paper.

A finite p-group G is q-self dual if every quotient of G is isomorphic to a subgroup of G. Here, we determine finite 2-groups G all of whose subgroups are q-self dual (Theorem 3) and in case p > 2 we get a classification of such groups only under the additional assumptions that Ω₁(G) is abelian (Theorem 4).

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ₂(Z) which is not clean.

In this paper, we give the definitions of almost cocommutative Hopf coquasigroup and quasitriangular Hopf coquasigroup which are analogous to those in the theory of Hopf algebras. Using the actions of Hopf coquasigroups, we obtain two equivalent conditions for a Hopf coquasigroup to be almost cocommutative. Furthermore, a quasitriangular Hopf coquasigroup determines a class of solutions of the quantum Yang–Baxter equation (QYBE). Finally, we define a 2-cocycle on a Hopf coquasigroup and construct a triangular Hopf coquasigroup by using a Hopf coquasigroup with a 2-cocycle.

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of the Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter ℤ²-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra gl(n | m). Then we extend the result to two-parameter ℤ-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras osp(2n | 2m) and osp(2n + 1 | 2m).

In this paper, we define and characterize the f-graphs. Also, we give a construction of f-graphs and importantly we show that the f-graphs obtained from this construction are Cohen–Macaulay.

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

We consider two natural concepts of "perfectness" in a semiring context: the perfectness in terms of projective covers, and the perfectness in Bass's sense. For both cases we show that the class of right (left) perfect semirings coincides with the class of right (left) perfect rings; in particular, we completely confirm Conjecture of [Y. Katsov, Toward homological classification of semirings: Serre's conjecture and Bass's perfectness in a semiring context. Algebra Universalis52 (2004) 197–214].

In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; we also describe congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.

In this paper, we mainly consider some special Ore extension of quasitriangular Hopf group coalgebra, and give the necessary and sufficient conditions when the Ore extension of quasitriangular Hopf group coalgebras will preserve the same quasitriangular structure. Furthermore, in the two examples given at the end, we construct new solutions of Yang–Baxter equation of Hopf group coalgebras version.

The purpose of this paper is to provide an octonionic description of the Lie group SL(2, 𝕆). The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from 𝔥₂(𝕆) onto itself. An interesting characterization is given for the generators of G₂. Also, explicit isomorphisms are constructed between the Lie algebras 𝔰𝔩(2, 𝕂), for 𝕂 = ℝ, ℂ, ℍ, 𝕆, and their corresponding Lorentz algebras.

In this paper, we investigate the transfer of the notions of pm-rings, rings of finite character and h-local rings to trivial ring extensions of rings by modules, amalgamations of rings along ideals and pullbacks. Our aim is to provide new classes of commutative rings satisfying these properties and our results generate new families of examples of rings for which the finite character and semi-local properties are equivalents.

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring R, then for any element a ∈ R and central units u₁, u₂, …, u_n ∈ U(R) there exists a unit u ∈ U(R) such that a + u_iu ∈ U(R) for each i ≥ 1.

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. The pseudo Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness) was introduced by Cuong–Nhan [Pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules, J. Algebra267 (2003) 156–177] as an extension of the Cohen–Macaulayness (respectively, generalized Cohen–Macaulayness). In this paper, we describe the pseudo Cohen–Macaulay (pseudo CM) locus and pseudo generalized Cohen–Macaulay (pseudo generalized CM) locus of M. We also study the non-Cohen–Macaulay locus and the non-generalized Cohen–Macaulay locus of the canonical module K(M) of M in case where R is a quotient of a Gorenstein local ring.

Over an artin algebra Λ, for an additive subbifunctor F of with enough projectives and injectives, we introduce F-Gorenstein dimensions in this paper. The new relative dimensions are useful to characterize F-Gorenstein algebras and F-self-injective algebras. In addition, with the aid of F-Gorenstein dimensions, we obtain an equivalent condition for the finitistic dimension conjecture to hold, that is, fin.dim Λ < ∞ for all artin algebras Λ if and only if rel.fin.Gdim_F Λ < ∞ for all artin algebras Λ.

A ring R is called right (small) dual if every (small) right ideal of R is a right annihilator. Left (small) dual rings can be defined similarly. And a ring R is called (small) dual if R is left and right (small) dual. It is proved that R is a dual ring if and only if R is a semilocal and small dual ring. Several known results are generalized and properties of small dual rings are explored. As applications, some characterizations of QF rings are obtained through small dualities of rings.

We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

This paper is a contribution to the investigation of flag-transitive primitive symmetric (v, k, λ) designs with λ at most 10. We prove that if 𝒟 = (𝒫, ℬ) is a nontrivial symmetric (v, k, λ) design, G ≤ Aut(𝒟) is flag-transitive primitive with Soc(G) = A_n for n ≥ 5, then 𝒟 is the unique symmetric (35,18,9) design which is the complement of the symmetric (35,17,8) design.