Journal of Algebra and Its Applications

Let 𝔉 be a class of finite groups. A subgroup H of G is said to be 𝔉-supplemented in G if there exists a subgroup T of G such that G = TH and (H ∩ T)H_G/H_G is contained in the 𝔉-hypercenter of G/H_G. In this paper, we investigate further the influence of 𝔉-supplemented subgroup on the structure of finite groups and generalize a series of known results.

We describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra over a field of characteristic p≠2 (infinite field in the case of the unitary algebra).

We find relations for finite-dimensional irreducible representations of the simple Lie algebra 𝔤₂, resulting in an explicit construction of Gröbner–Shirshov pairs and monomial bases for 𝔤₂-modules V(nΛ₂).

Derivations of the ring of all finitary niltriangular matrices over an arbitrary associative ring with identity for any chain of matrix indices are described. Every Lie or Jordan derivation is a derivation of this ring modulo third hypercenter.

We first construct algebra automorphisms T_i (1 ≤ i ≤ r) of ℤ₂-graded Hopf algebra U_q(osp(1|2r)), which are called Lusztig's symmetries. It is shown that they satisfy braid group relations. The algebraic automorphisms of U_q(osp(1|2r)) are also described.

We prove that the set of right 4-Engel elements of a group G is a subgroup for locally nilpotent groups G without elements of orders 2, 3 or 5; and in this case the normal closure ⟨x⟩^G is nilpotent of class at most 7 for each right 4-Engel elements x of G.

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.

Let D be an integral domain. We investigate two invariants ω(D, x) and ω(D) which measure how far an x ∈ D is from being prime and how far an atomic integral domain D is from being a UFD, respectively. We give a new characterization of number fields with class number two. We also study asymptotic versions of these two invariants.

We classify all unital subalgebras of the Cayley algebra 𝕆(q) over the finite field F_q, q = pⁿ. We obtain the number of subalgebras of each type and prove that all isomorphic subalgebras are conjugate with respect to the automorphism group of 𝕆(q). We also determine the structure of the Moufang loops associated with each subalgebra of 𝕆(q).

Let R be a prime ring with extended centroid C, a nonzero ideal I and two derivations d₁, d₂. Suppose that d₁(x)ⁿ = d₂(x)ⁿ for all x ∈ I. Then there exists λ ∈ C such that d₂(x) = λd₁(x) for all x ∈ R.

Let p be a prime number. A formula for the minimum absolute value of the discriminant of all Abelian extensions of ℚ of degree p² is given in terms of p.

Let M be a Krull monoid. Then every element of M may be written as a finite product of irreducible elements. If for every a ∈ M, each two factorizations of a have the same number of irreducible elements, then M is called half-factorial. Using a property of element exponentiation, we provide a new characterization of half-factoriality, valid for all Krull monoids whose class group has torsion-free rank at most one.