Journal of Algebra and Its Applications

In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M₂(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

Let R be a prime ring that is not commutative and such that R ≇ M₂(GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x^m)xⁿ = xⁿG(x^m) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x^m and xⁿ are C-dependent for all x ∈ R. We also consider the situation for the semiprime case.

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim(A ⊗_k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain.

The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if G₁ and G₂ have the same configuration sets and H₁ is a normal subgroup of G₁ with abelian quotient, then there is a normal subgroup H₂ of G₂ such that . Also configuration of FC-groups and isomorphism is studied.

We introduce the notion of classical primary submodules that generalizes the concept of primary ideals of commutative rings to modules. Existence and uniqueness of classical primary decompositions in finitely generated modules over one-dimensional Noetherian domains are proved.

Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

A right R-module M_R is called small injective if every homomorphism from a small right ideal to M_R can be extended to an R-homomorphism from R_R to M_R. A ring R is called right small injective, if the right R-module R_R is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module E_R, E^(ℕ) is small injective.

Based on the prime graph of a finite group, its order can be divided into a product of some co-prime positive integers. These integers are called order components of this group. If there exist exactly k nonisomorphic finite groups with the same set of order components of a given finite group, we say that it is a k-recognizable group by its order component(s). In the present paper, we obtain that all finite simple K_n-groups (n = 3, 4) except U₄(2) and A₁₀ can be uniquely determined by their order components. Moreover, U₄(2) is 2-recognizable and A₁₀ is k-recognizable, where k denotes the number of all nonisomorphic classes of groups with the same order as A₁₀. As a consequence of this result we can obtain some interesting corollaries.

The existence of loop rings that are not associative but which satisfy the Moufang or Bol identities is well known. Here we complete work started 25 years ago by establishing the existence of loop rings that satisfy any identity of "Bol–Moufang" type (without being associative). As it turns out, with one exception, loop rings satisfying an identity of Bol–Moufang type all satisfy a Moufang or Bol identity. We also highlight some similarities and differences in the consequences of several Bol–Moufang identities as they apply to loops and rings.

In this paper we prove that the (non-trivial) irreducible complex unipotent characters of a finite classical group G of type B_m or C_m in odd characteristic are reducible over any proper subgroup of G, apart from very few notable exceptions.