Journal of Algebra and Its Applications

In this paper, we give a sufficient (which is also necessary under a compatibility hypothesis) condition on a set of arrows in the quiver of an algebra A so that A is a split extension of A/M, where M is the ideal of A generated by the classes of these arrows. We also compare the notion of split extension with that of semiconvex extension of algebras.

A differential quasifield is a natural generalization of a differential field in characteristic p > 0. Elementary properties of differential quasifields are considered, and a generalized version of the theorem on the connection between linear independence over constants and the wronskian is presented. The notion of a partitioned differential quasifield is introduced, and the behavior of partitioned differential quasifields is considered, including under an extension of scalars.

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal cover with group the fundamental group of the underlying space of Q.

A semidirect product is introduced for cycloids, i.e. sets with a binary operation satisfying (x · y) · (x · z) = (y · x) · (y · z). Special classes of cycloids arise in the combinatorial theory of the quantum Yang–Baxter equation, and in algebraic logic. In the first instance, semidirect products can be used to construct new solutions of the quantum Yang–Baxter equation, while in algebraic logic, they lead to a characterization of L-algebras satisfying a general Glivenko type theorem.

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E₁ → E₀ → E⁰ → E¹ → ⋯ of injective R-modules with M = ker(E⁰ → E¹) and such that Hom(E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended.

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

We classify simple finite Jordan conformal superalgebras and also establish preliminary results for the classification of simple finite Jordan pseudoalgebras.