Formal Triangular Matrix Rings And The Modules Over Them

Formal triangular matrix rings are an important class of noncommutative rings, they play a vital role not only in classical theory of rings and modules, but also in representation theory of Artin algebras. With their asymmetric structures, they provide a good source of examples and counterexamples in ring theory.Many ring-theoretic properties (e.g., CS property, quasi-Baer property, (strong) FI-exten-ding property, perfectness, semiperfectness, self-injectiveness, IBN property, etc.), as well as the (Jordan) isomorphism and (Jordan) derivation, and the homological properties of modules over them (e.g., projective module and dual basis, projective generator, projective cover, injet-ive module, injective cogenerator, injective hull, flat module, etc.) have been specially studied. Furthermore, a lot of similar research has been done on a wider class of ring: ring of Morita context. However, it can't be denied that plenty of new properties remain to be investigated.In this paper,some new ring-theoretic properties are studied: right principally quasi-Baer property, right PCS-property and right strongly PCS-property. Equivalent characterizations are given for the right principally quasi-Baer property and the right strongly PCS-property. It's shown that the upper triangular matrix ring over a right principally quasi-Baer ring (or a right strongly PCS-ring) is also right principally quasi-Baer (or right strongly PCS). We also give a sufficient condition for a formal triangular matrix ring to be a right PCS-ring.Relative projectivity and relative injectivity in the module category over formal triangular matrix rings are also studied. Correspond to the concepts of simple injectivity, simple reject and simple trace, some new definitions are given, namely maximal projectivity, maximal reject and maximal trace. Given two right modules V₁ and V₂ over T, sufficient conditions are obtained for V₁ to be V₂ -projective. Relative maximal projectivity is characterized using maximal reject and maximal trace. With the concept of generalized trace: a generalization of trace, we also give a sufficient condition for Y₁ to be Y₂ -injective when V₁ is V₂ - injective.