One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summa...One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.展开更多
A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copre...A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copres∞（U）. A module U is a co-＊∞-module if and only if U is co-selfsmall and for any exact sequence 0→M→UI→N→0 with M∈Copres∞（U） and I is a set, N∈Copres∞（U） is equivalent to Ext1R（N,U）→Ext1R（UI,U） is a monomorphism if and only if U is co-selfsmall and for any exact sequence 0→L→M→N→0 with L, N∈Copres∞（U）, N∈Copres∞（U） is equivalent to the induced sequence 0→Δ（N）→Δ（M）→Δ（L）→0 which is exact if and only if U induces a duality ΔUS：⊥USCopres∞（U）：ΔRU. Moreover, U is a co-＊n-module if and only if U is a co-＊∞-module and Copres∞（U）=Copresn（U）.展开更多
Some decompositions of modules over a ring can be discussed in special categories Here we study the decomposition of quasi-injective objects over a Grothendieck category and generalize some propositions on quasi-injec...Some decompositions of modules over a ring can be discussed in special categories Here we study the decomposition of quasi-injective objects over a Grothendieck category and generalize some propositions on quasi-injective modules over a ring in [1] and [3].展开更多
Let R be a ring. R is called right AP-injective if, for any a E R, there exists a left ideal of R such that lr（a） = Ra＋Xa. We extend this notion to modules. A right R-module M with S = End（MR） is called quasi AP-...Let R be a ring. R is called right AP-injective if, for any a E R, there exists a left ideal of R such that lr（a） = Ra＋Xa. We extend this notion to modules. A right R-module M with S = End（MR） is called quasi AP-injective if, for any s∈S, there exists a left ideal Xs of S such that ls（Ker（s）） = Ss＋Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.展开更多
文摘One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.
基金The National Natural Science Foundation of China (No.10971024)Specialized Research Fund for the Doctoral Program of Higher Education (No.200802860024)
文摘A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copres∞（U）. A module U is a co-＊∞-module if and only if U is co-selfsmall and for any exact sequence 0→M→UI→N→0 with M∈Copres∞（U） and I is a set, N∈Copres∞（U） is equivalent to Ext1R（N,U）→Ext1R（UI,U） is a monomorphism if and only if U is co-selfsmall and for any exact sequence 0→L→M→N→0 with L, N∈Copres∞（U）, N∈Copres∞（U） is equivalent to the induced sequence 0→Δ（N）→Δ（M）→Δ（L）→0 which is exact if and only if U induces a duality ΔUS：⊥USCopres∞（U）：ΔRU. Moreover, U is a co-＊n-module if and only if U is a co-＊∞-module and Copres∞（U）=Copresn（U）.
文摘Some decompositions of modules over a ring can be discussed in special categories Here we study the decomposition of quasi-injective objects over a Grothendieck category and generalize some propositions on quasi-injective modules over a ring in [1] and [3].
文摘Let R be a ring. R is called right AP-injective if, for any a E R, there exists a left ideal of R such that lr（a） = Ra＋Xa. We extend this notion to modules. A right R-module M with S = End（MR） is called quasi AP-injective if, for any s∈S, there exists a left ideal Xs of S such that ls（Ker（s）） = Ss＋Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.
