In the paper derivations and centroids of four-dimensional associative algebras are described. We also identify the class of algebras called characteristically nilpotent among four-dimensional associative algebras.

In this study, we extend the result on classification of a subclass of filiform Leibniz algebras in low dimensions to dimensions seven and eight based on the technique used by Rakhimov and Bekbaev for classification of subclasses which arise from naturally graded non-Lie filiform Leibniz algebras. The subclass considered here arises from naturally graded filiform Lie algebras. This subclass contains the class of filiform Lie algebras and consequently, by classifying this subclass, we again re-examine the classification of filiform Lie algebras. Our resulting list of filifom Lie algebras is compared with that given by Ancochéa-Bermudez and Goze in 1988 and by Gómez, Jimenez-Merchan and Khakimdjanov in 1998.

In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed.

Abstract. This paper gives generalization of a notion of supplemented module. Here, we utilize some algebraic properties like supplemented, amply supplemented and local modules in order to obtain the generalization. Other properties that are instrumental in this generalization are D i , SSP and SIP . If a module M is Rad-D 11 -module and has D 3 property, then M is said to be completely-Rad-D 11 -module (C-Rad-D 11 -module). Similarly it is for M with SSP property. We provide some conditions for a supplemented module to be C-Rad-D 11 -module. Introduction and PreliminariesThroughout this paper all rings are unital and modules are considered to be right

No abstract