Cotorsion pairs and approximation classes over formal triangular matrix rings

Abstract

Let T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Let C1 and C2 be two classes of left A-modules, D1 and D2 be two classes of left B-modules, we prove that: (1) If ToriA(U,C1)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (PD1C1,AD2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. (2) If ExtBi(U,D2)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (AD1C1,ID2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. In addition, we characterize some special preenveloping classes and precovering classes in T-Mod.