The Relative Properties Of Closed Ideals

In the theory of modules,the direct sum decomposition of modules is one of its central problems,in which the definition of closed submodule as the extension of that direct summands is a very basic and important concept in the theory of rings and modules.For example,Goldie first introduced the theory of uniform dimensions of modules,the notion of a closed submodule plays a crucial role in this theory.In 1998,Santa-Clara gave an equivalent characterization of quasi-continuous modules:M is quasi-continuous if and only if the direct sum of any two closed submodules in M is still its closed submodule.So,the idea of CSP(the closed sum property)modules came:M is called a CSP module if the sum of any two closed submodules of M is again closed.And in the theory of rings,a closed right(left)ideal of a ring is a closed submodule of RR(RR)Inspired by CSP modules,in this dissertation,CSP rings are defined and properties of this new class of rings are explored.If R is right CS,each closed right ideal of R is a direct summand of RR.Because each direct summand of RR can be written as eR,where e is an idempotent of R.In fact,eR is cyclic.It is natural to consider the rings whose closed right ideals are cyclic.Therefore,CC rings are defined and properties of this new class of rings are explored.This paper will focus on the relative properties of closed ideals in rings.It mainly consists of the following three partsThe first part mainly focuses on the CSP rings.Firstly,it is proved that R is a right CSP ring if and only if R is a right CS and SSP ring.And an example of a left CSP ring which is not right CSP is given.Next,it is proved that the matrix ring Mn(R)over R is a right CSP ring if and only if R is regular and right self-injective.The equivalent conditions for a column finite matrix ring CFIM?(R)over R to be a right CSP ring are given.Finally,the CSP of the trivial extension rings is explored.Incidentally,the following interesting result is found:R ? R is an SSP ring if and only if R is an abelian ringThe second part mainly considers the closed ideals of 2 x 2 triangular matrix rings Although through the study of CSP of triangular matrix rings,triangular matrix rings are not CSP rings.But the closed ideals of upper triangular matrix rings over R are studied A necessary and insufficient condition is obtained.A counter example is given to explain the insufficiency.The equivalence forms of special closed ideals are also described.And the relative results are extended to any n x n upper triangular matrix rings Tn(R)The third part mainly deals with CC rings.Through the definition of CC rings,CS rings are right CC rings.An example is given to show that right CC rings may not be right CS.So the class of right CC rings is a real generalization of the class of right CS rings.It is proved that if R is a right 2-injective ring,then R is right CS if and only if R is right CC.Finally,the CC properties of the matrix ring Mn(R)and the column finite matrix ring CFIM?(R)over R are discussed.Cyclic ideals of trivial extension rings and upper triangular matrix rings over R are described.And their CC properties are studied.