Study Of Modules,Maps And Zero-divisor Graphs Over Formal Matrix Rings

Formal matrix rings generalize the matrix rings over a given ring, which play an important role in ring theory and module theory. It is well known that every ring with nontrivial idempotents is isomorphic to a formal matrix ring, and the endomorphism ring of a decomposable module is also isomorphic to some formal matrix ring. The class of formal triangular matrix rings is an important class of formal matrix rings,which plays an important role in the representation theory of Artinian algebras.Formal matrix rings have abundant properties and important applications. In this thesis, we study modules, maps and zero-divisor graphs of formal matrix rings.The first chapter introduces the research background of this thesis and relative basic concepts and results.The second chapter is mainly on the artinian, noetherian and finitely presented modules over a formal matrix ring ???. We obtain that any right K -module?X,Y?f,g is right artinian ?noetherian? if and only if so are X and Y . We give a sufficient condition for a right K -module to be finite presented module. Moreover, we also study the superfluous covers of a right K -module and show that every right K -module ?X, Y?f,g has a projective cover if and only R -module X / YN and S -module Y/XM have projective covers.The third chapter focus on the homomorphisms, ?-derivations, ?-biderivations and a -commuting mappings of formal matrix rings with zero trace ideals. It is proved that under certain conditions every ? -biderivation of a formal matrix ring is a sum of an inner a -biderivation and an extremal ? -biderivation and give a sufficient condition for every ? -biderivation to be inner ? -biderivation. Furthermore. We describe the general form of a -commuting mappings of formal matrix rings, and obtain several equivalent characterizations of proper ? -commuting mappings and give a sufficient condition for a ? -commuting mapping to be proper.The fourth chapter is devoted to the zero-divisors and zero-divisor graph of formal matrix ring M_n(R;{S_ijk}) over a commutative ring R. We first introduce the notion of left ?right? system of formal linear equations over a ring R , and apply it to prove that an element A of M_n(R;{S_ijk}) is a zero-divisor if and only if its determinant is a zero-divisor in R if and only if A is a zero-divisor of R[A]. Then we investigate the undirected zero-divisor graph ? (M_n(R;{S_ijk})) and the directed zero-divisor graph?(M_n(R;{S_ijk})) of the formal matrix ring M_n(R;{S_ijk}). It is showed that for any integer n>1, ?(M_n(R;{S_ijk}))is not planar, the girth of ?(M_n(R;{S_ijk}))is 3, and the diameter of ?(M_n(R;{S_ijk})) is 2 or 3. Finally, we also proved that the diameter of?(M_n(R;{S_ijk}))is 2 or 3 and ?(M_n(R;{S_ijk})) ??? ?(M_n(T?R?;{S_ijk})), where T?R? is the total quotient ring of R .