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.

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).

Let R be a non-necessarily commutative ring and M an R-module. We use the category σ[M] to introduce the notion of I-module who is a generalization of I-ring. It is well known that every artinian object of σ[M] is cohopfian but the converse is not true in general.The aim of this paper is to characterize for a fixed ring, the left (right) R-modules M for which every co-hopfian object of σ[M] is artinian.We obtain some characterization of finitely generated I-modules over a commutative ring, faithfully balanced finitely generated I-modules, and left serial finitely generated I-modules over a duo-ring.