We show that every 4-dimensional real division algebra having non-zero derivations is obtained from the real algebra C by a special kind of duplication process accompanied by an appropriate isotopy. Also this process produces new examples in dimension 8.

We study the absolute valued algebras containing a central element non necessary idempotent. We determine the absolute valued algebras containing a central element if we add some requirements. Also we gives a classification of finitedimensional absolute valued algebras containing a generalized left unit and central element.

We characterize the non-triviality of the group of automorphisms for all absolutevalued algebras of finite dimension ≥ 4.

Let R be a commutative ring, with a unity 1 0 and M a unitary left R-module. In this paper we give some properties of an FGS -module. After that we give others important characterizations. Indeed, we first show that M is a local FGS -module if and only if it is of finite representation type. Secondly, we show that M is a prime FGS -module if and only if it is a serial type module and of finite length if and only if it is a finite representation type module.

In this paper, we study partially the automorphisms groups of four-dimensional division algebra. We have proved that there is an equivalence between Der(A) = su(2) and Aut(A) = S O(3). For an unitary four-dimensional real division algebra, there is an equivalence between dim(Der(A)) = 1 and Aut(A) = S O(2).