Formal Triangular Matrix Rings And Some Modules Over It

This paper mainly studies the properties of formal triangular matrix rings and characterize some special modules over it. Most of results are well known.In the introduction part, we recall the development of history and the background of the formal of triangular matrix rings. And we also introduce some preliminaries.In Chapter2, we discuss the submodules of modules over a formal triangular matrix ring. Firstly we give some equivalent characterization of the scole of a modules over a formal triangular matrix rings. Secondly, we discuss the maximal proper submodule, minimal non-zero submodule and small submodule of a module over a formal triangular matrix ring respectively, And get some equivalent characterization of them.In Chapter3, we summary some important properties of the formal triangular matrix rings, such as semi-completeness、completeness, and get some necessary and sufficient conditions of a formal triangular matrix ring is V-rings and semi-V-ring respectively. Finally, we obtain some the equivalent characterization of a formal triangular is a Max--ring.Chapter4deals with some special modules over a formal of triangular matrix rings. Firstly, we give some equivalent conditions of a module over a formal of triangular matrix ring is noetherian. Secondly, we discuss some properties of P-projective modules over a formal triangular matrix ring. At last, we give an equivalent characterization of a module over a formal triangular matrix ring is a P.P. module.