We have this two Lemma
Lemma 1: Let $R$ be a commutative principal ideal ring and $M$ be a free left module of finite rank $l$ over $R$. Then every submodule $N$ of $M$ can be generated by at most $l$ elements.
Lemma 2: Let $l$ be a positive integer and $R$ be a commutative principal ideal ring. Then there is a one-to-one correspondence between the submodules of $R^{l}$ and the left ideals of $M_{l}(R)$.
Now in the proof of Lemma 2. Let N ⊆ $R^{l}$ , we can build a left ideal of M_{l}(R) whose elements have rows in N. Conversely, to a left ideal I ⊆ M_{l}(R) we associate the submodule of $R^{l}$ generated by all the rows of all the elements of I . It is straightforward to check that these maps are inverse to each other.
Now if we take $R=GF(q)[X]/(X^{m}-1)$
It's easy to verify that there's a one to one correspondence between $l$-quasi-cyclic codes over $GF(q)$ of length $ml$ and sub-modules of $(GF(q)[X]/(X^{m}-1))^l$
My question is what's the construction of the left ideal $M_{l}(R)$? and what does it mean by saying whose elements have rows in N? and how we can apply this in the case of $l$-quasi-cyclic codes over $GF(q)$ of length $ml$ to have correspondence with $M_{l}(R)$?