Almost Strongly Presented Modules And Generalized Strongly Presented Modules

We define and investigate almost strongly presented modules as a special class of almost finitely presented modules. A module M is called almost strongly presented module in case M=M’(?) M*, where M’ is a strongly presented module and M* is an infinitely generated free module. The class of almost strongly presented modules is closed under direct sum. Many equivalent conditions for a module M to be almost strongly presented module are given. We also define and investigate generalized strongly presented modules as a special class of generalized finitely presented modules. A module M is called generalized strongly presented module in case M≌P/A, where P is a project module and A is a strongly presented module, that is, there is an exact sequence Oâ†'Aâ†'Pâ†'Mâ†'O. And get the structure theorem of generalized strongly presented modules. Finally we deal with the dual modules of generalized strongly presented modules and the connection between the generalized strongly presented modules and almost strongly presented modules. It is organized as follows:In chapter 1, we introduce the background and development of almost strongly presented modules and generalized strongly presented modules, and list some basic conceptions and lemmas which are used in this dissertation.In chapter 2, we introduce the notion of almost strongly presented modules. We will show that almost strongly presented modules admit many other characterizations. The direct sum of almost strongly presented modules is also an almost strongly presented module.The following are the main results of this chapter:Theorem 2.5 Let M be an R-module, then M is an almost strongly presented module if and only if there exact sequence where F0 is an infinitely generated free module, Fi (i=1,2,…,m,…) are finitely generated projective modules.Corollary 2.6 Let M be an R-module, then M is an almost strongly presented modules if and only if there exact sequence where F1 is a strongly presented module, F is an infinitely generated free module.Theorem 2.7 Assume that Oâ†'Aâ†'Bâ†'Câ†'O is an exact sequence of left F-modules, A, C are almost strongly presented modules, then so is B.Corollary 2.8 Let M1, M2, …, Mn are left R-modules, M1, M2, …, Mn are almost strongly presented modules, then so is (?)in=1Mn.Proposition 2.12 Let M be an almost strongly presented R-module, then Fpd(M)=1.In chapter 3, we introduce the concept of generalized strongly presented modules and get their structure theorem, and discuss the dual modules of generalized strongly presented modules. Finally the relations between the generalized strongly presented modules and almost strongly presented modules are studied.The following are the main results of this chapter:Theorem 3.1.3 Let M be an R-module, M is a generalized strongly presented module if and only if there are a projective module P0, a free module F*, and a strongly module M0 such that M(?)P0=M0(?) F*Theorem 3.2.2 Assume that R is a ring such that every finitely generated submodule of projective modules is strongly presented, let M be a strongly presented module, then M*= Hom(M, R) and ExtRn(M, R) are strongly presented modules.Theorem 3.2.3 Assume that R is a ring such that every finitely generated submodule of projective modules is strongly presented, let M be a generalized strongly presented module, then ExtRn[M, R)(n≥ 1) are strongly presented.Theorem 3.3.2 Let M be an R-module, M is generalized strongly presented if and only if there are a projective module P0, an infinitely generated free module F*, and a strongly module M0 such that M(?)P0=M0(?)F*.Corollary 3.3.3 Let M be an R-module, M is generalized strongly presented if and only if there are a projective module Po such that M(?)P0 is almost strongly presented.Theorem 3.3.6 The following statements are equivalent for a ring R:(1) Every strongly presented submodule of projective R-modules is projective, and every strongly presented modules is projective;(2) Every almost strongly presented R-module is projective;(3) The direct summand of almost strongly presented R-modules are projective;(4) The generalized strongly presented direct summand of almost strongly presented R-modules are projective.