Some Classes of Nilpotent Associative Algebras

We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.     

Classification of solvable Leibniz algebras with null-filiform nilradical

In this article we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra as well.     

Description of some classes of Leibniz algebras

In this paper we describe the isomorphism classes of finite-dimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl₂ and a three-dimensional solvable ideal. We choose a basis of the isomorphism classes’ representatives and give explicit multiplication tables.