The phenomenon of defectless extensions is a classical notion in the framework of valued  elds and valued vector spaces in valuation theory. The aim of this paper is to study various results regarding this concept and its applications.

Let R be an associative ring with identity. An element x 2 R is called Z G -regular (resp. strongly ZG -regular) if there exist g ÎG, nÎZ and rÎR such that xng=xngrxng (resp. xng=x (n+1) g).A ringR is called Z G -regular (resp. strongly Z G -regular) if every element of R is Z G -regular (resp.stronglyZ G -regular). In this paper, we characterize Z G -regular (resp. strongly Z G -regular) rings.Furthermore, this paper includes a brief discussion of Z G -regularity in group rings.

Let A be a unital R-algebra and M be a unital A- bimodule. It is shown that every Jordan derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

The aim of this paper is to study the (a; g) -prolongation of central extensions. We obtain the obstruction theory for (a; g) -prolongations and classify (a; g) -prolongations thanks to low-dimensional cohomology groups of groups.