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
Let R be a ring. A right R-module M with S = End（MR） is called a quasi AP-injective module, if, for any s C S, there exists a left ideal Xs of S such that ls（ker s） = Ss＋Xs. Let M be a quasi AP-injective module w...Let R be a ring. A right R-module M with S = End（MR） is called a quasi AP-injective module, if, for any s C S, there exists a left ideal Xs of S such that ls（ker s） = Ss＋Xs. Let M be a quasi AP-injective module which is a self-generator. We show that for such a module, if S is semiprime, then every maximal kernel of S is a direct summand of M. Furthermore, if ker（a1） lohtain in ker（a2a1） lohtain in ker（a3a2a1） lohtain in... satisfy the ascending conditions for any sequence al, a2, a3,… ∈ S, then S is right perfect. In this paper, we give a series of results which extend and generalize results on AP-injective rings.展开更多
The purpose of this paper is to study the following two questions on AP-injective rings: (1) R is a regular ring if and only if R is a left PP-ring and R is left AP-injective; (2) Let R be a right .AP-injective ring. ...The purpose of this paper is to study the following two questions on AP-injective rings: (1) R is a regular ring if and only if R is a left PP-ring and R is left AP-injective; (2) Let R be a right .AP-injective ring. Then R is self-injective if and only if R is weakly injective. Hence we get some new results of P-injective rings.展开更多
Some relations about the generalizations of self-injective ring: P-injective ring, GP-injective ring, AP-injective ring, simple-injective ring and n-injective ring are studied.
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.
文摘Let R be a ring. A right R-module M with S = End（MR） is called a quasi AP-injective module, if, for any s C S, there exists a left ideal Xs of S such that ls（ker s） = Ss＋Xs. Let M be a quasi AP-injective module which is a self-generator. We show that for such a module, if S is semiprime, then every maximal kernel of S is a direct summand of M. Furthermore, if ker（a1） lohtain in ker（a2a1） lohtain in ker（a3a2a1） lohtain in... satisfy the ascending conditions for any sequence al, a2, a3,… ∈ S, then S is right perfect. In this paper, we give a series of results which extend and generalize results on AP-injective rings.
基金Supported by the NNSF of China(10071035)the Foundation of the Education Committee of Anhui Province(2003kj166).
文摘The purpose of this paper is to study the following two questions on AP-injective rings: (1) R is a regular ring if and only if R is a left PP-ring and R is left AP-injective; (2) Let R be a right .AP-injective ring. Then R is self-injective if and only if R is weakly injective. Hence we get some new results of P-injective rings.
基金Supported by National Natural Science Foundation of China(10071062)
文摘Some relations about the generalizations of self-injective ring: P-injective ring, GP-injective ring, AP-injective ring, simple-injective ring and n-injective ring are studied.
