  Let $ K / k $ be a purely inseparable extension of characteristic $ p> 0 $
and of finite size. We recall that $K/k$ is modular if for every $n \in
\mathbb{N}$,$K^{p^n}$ and $k$ are $k\cap K^{p^ n}$-linearly disjoint. A natural
generalization of this notion is to say that $K/k$ is $lq$-modular if $K$ is
modular over a finite extension of $k$. Our main objective is to extend in
definite form the results and definitions of the $lq$-modularity that have
already been obtained in the case limited by the finiteness condition imposed
on $[k :k^p]$ in a rather general framework (framework of extensions of finite
size called also $q$-finite extensions).First, by means of invariants, we
characterize the $lq$-modularity of a $q$-finite extension. Next, we show that
any intersection of a $q$-finite extensions covering $k$ or $K$ preserves the
$lq$-modularity. We also prove that any $q$-finite extension $ K/k $ contains a
greater $lq$-modular and relatively perfect sub-extension. In particular, this
result is very useful for defining the modularity of order $i$ linked to a
$q$-finite extension $ K/k $.Moreover, we give a necessary and sufficient
condition for $K/k$ to be $i$-modular. Certainly, the modularity level of $ K /
k $ never exceeds the sizeof $K/k$. Notably, we explicitly describe the
extension $K/k $ whose degree of modularity is the size of $ K / k $. In the
end, we examine a particular decomposition of $ K/k $ defined by inverse
chaining.
