  Generalizing the linear complementary duals, the linear complementary pairs
and the hull of codes, we introduce the concept of $\ell$-dimension linear
intersection pairs ($\ell$-DLIPs) of codes over a finite commutative ring
$(R)$, for some positive integer $\ell$. In this paper, we study $\ell$-DLIP of
codes over $R$ in a very general setting by a uniform method. Besides, we
provide a necessary and sufficient condition for the existence of a non-free
(or free) $\ell$-DLIP of codes over a finite commutative Frobenius ring. In
addition, we obtain a generator set of the intersection of two constacyclic
codes over a finite chain ring, which helps us to get an important
characterization of $\ell$-DLIP of constacyclic codes. Finally, the $\ell$-DLIP
of constacyclic codes over a finite chain ring are used to construct new
entanglement-assisted quantum error correcting (EAQEC) codes.
