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.

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