Extension of i-modularity

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.