Journal of Algebra and Its Applications

In this paper, we study on Moufang–Klingenberg planes defined over a local alternative ring of dual numbers. We get relations between some special 6-figures and its ratios. Also, we construct a relation between the solvability of the special quadric equation in and the existence of the special 6-figure in . Moreover, we give a construction which provides an algorithm for "squaring" and "cubing" a cross-ratio of points in .

An R-module _RM is called morphic if M/im α ≅ ker α for every endomorphism α of M, that is, if the dual of the Noether isomorphism theorem holds. Mostly all morphic Z-modules are determined leaving open some classes of nonsplitting mixed groups, which actually cannot be completely characterized. However, for these classes, comprehensive information including negative examples is given.

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽_q) and PSL(2, q) = PSL(2, 𝔽_q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups.

As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M₂₀ or M₂₁ = PSL(3, 4) is the group algebra.

A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image.

In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Gröbner–Shirshov bases of free λ-differential Rota–Baxter algebras. In particular, linear bases of free λ-differential Rota–Baxter algebras are obtained and consequently, the free λ-differential Rota–Baxter algebras are constructed by words.

The most important families of nonlinear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper, we provide a Gröbner basis technique to compute the weight/distance distribution of any systematic nonlinear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families.

A classification of Lie solvable group algebras of derived length 3 over fields of characteristic 3 and 5 is given. It is known that for a field K with Char K ≥ 7 and G an arbitrary group, δ^[3](L(KG)) = 0 if and only if δ⁽³⁾(KG) = 0. Same turns out to be true when Char K = 5 and G is an arbitrary group or when Char K = 3 and G is a two-Engel group.

A ring R is called left morphic if, for each a ∈ R, R/Ra ≅ l(a) or equivalently there exists b ∈ R such that Ra = l(b) and l(a) = Rb, where l(a) and l(b) denote the left annihilators of a and b in R, respectively. Motivated by recent work on left morphic rings, we study the rings R satisfying the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b) and l(a) = Rb. These rings are completely determined in this paper. We also completely determine the rings R with the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b).

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

Let K be a field of characteristic p ≠ 2,3 and let G be a finite group. Necessary and sufficient conditions for δ₃(U(KG)) = 1, where U(KG) is the unit group of the group algebra KG, are obtained.

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth(𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of with dim N ≤ 1 the associated primes of are finite, whenever for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, for all i < l, and then there exists a generalized regular sequence x₁, …, x_l ∈ 𝔞 on M such that .

In this note, we shall study the Witten multiple zeta function associated with the exceptional Lie algebra 𝔤₂. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multi-polylogs evaluated at 12th roots of unity, except for two irregular cases.