Cotorsion pairs over formal triangular matrix rings of order n

Let [Formula: see text] be a formal lower triangular matrix ring of order [Formula: see text], where [Formula: see text] is not less than 3. In order to consider how to construct cotorsion pairs over [Formula: see text], we give some isomorphisms of the homological functor [Formula: see text]. As an application, we find some lower bounds of projective and injective dimensions of a left [Formula: see text]-module. In the paper, we define two special classes of left [Formula: see text]-modules. Under some conditions, we characterize these special classes of left [Formula: see text]-modules and construct cotorsion pairs associated to them over [Formula: see text] by cotorsion pairs over internal rings of [Formula: see text]. Moreover, we discuss complete and hereditary cotorsion pairs over [Formula: see text] and internal rings of [Formula: see text].