In this paper we first give a procedure 
by which we generate a filter by a subset in a transitive $BE$-algebra, 
and give some characterizations of Noetherian and Artinian  $BE$-algebras. 
Next we give the construction of  quotient algebra $X/F$ of
a transitive $BE$-algebra $X$ via a filter $F$ of $X$. 
Finally we discuss properties 
of Noetherian (resp. Artinian) $BE$-algebras on 
homomorphisms and prove that 
let $X$ and $Y$ be transitive $BE$-algebras, 
a mapping $f:\ X\rightarrow Y$ be an epimorphism. If $X$ is Noetherian (resp. Artinian), 
then so does $Y$.
Conversely suppose that $Y$ and $Ker(f)$ (as a subalgebra of $X$) are 
Noetherian (resp. Artinian), then so does $X$.
Let $X$ be a transitive $BE$-algebra  and $F$ a filter 
of $X$. If $X$ is Noetherian (resp. Artinian), then so does the quotient algebra $X/F$.