In this paper,we study extended modules for a special class of Oreextensions.We will assume that R is a ring and A will denote the Ore extensionA:=R[X1,...,xn;σ]for whichσis an automorphism of R,xixj=xjxi and xir=σ...In this paper,we study extended modules for a special class of Oreextensions.We will assume that R is a ring and A will denote the Ore extensionA:=R[X1,...,xn;σ]for whichσis an automorphism of R,xixj=xjxi and xir=σ(r)xi,for every 1<i,j≤n.With some extra conditions over the ring R,wewill prove Vaserstein's,Quillen's patching,Horrocks',and Quillen-Suslin's theoremsfor this type of non-commutative rings.展开更多
A module Mis called generalized extending if for any submodule N of M, there is a direct summand K of M such that N≤K and KIN is singular. Any extending module and any singular module are generalized extending. Any h...A module Mis called generalized extending if for any submodule N of M, there is a direct summand K of M such that N≤K and KIN is singular. Any extending module and any singular module are generalized extending. Any homomorphic image of a generalized extending module is generalized extending. Any direct sum of a singular （uniform） module and a semi-simple module is generalized extending. A ring R is a right Co-H-ring if and only if all right R modules are generalized extending modules.展开更多
We say a left R-module M is ecg-extending if every essentially countably generated submodule of M is essential in a direct summand of M. After giving some basic properties of ecg-extending modules, we show that for a ...We say a left R-module M is ecg-extending if every essentially countably generated submodule of M is essential in a direct summand of M. After giving some basic properties of ecg-extending modules, we show that for a nonsingular ring R, all left R-modules are ecg- extending if and only if all left R-modules are extending. We also characterize noetherian rings and artinian semisimple rings via ecg-quasi-continuous modules.展开更多
基金the project New trends of non-commutative algebra and skew PBW extensions,HERMES CODE 26872,Universidad Nacional de ColombiaThe authors are grateful tothe editors and the referee for valuable suggestions and corrections.
文摘In this paper,we study extended modules for a special class of Oreextensions.We will assume that R is a ring and A will denote the Ore extensionA:=R[X1,...,xn;σ]for whichσis an automorphism of R,xixj=xjxi and xir=σ(r)xi,for every 1<i,j≤n.With some extra conditions over the ring R,wewill prove Vaserstein's,Quillen's patching,Horrocks',and Quillen-Suslin's theoremsfor this type of non-commutative rings.
基金Project (No. 102028) supported by the Natural Science Foundation of Zhejiang Province, China
文摘A module Mis called generalized extending if for any submodule N of M, there is a direct summand K of M such that N≤K and KIN is singular. Any extending module and any singular module are generalized extending. Any homomorphic image of a generalized extending module is generalized extending. Any direct sum of a singular （uniform） module and a semi-simple module is generalized extending. A ring R is a right Co-H-ring if and only if all right R modules are generalized extending modules.
基金the National Natural Science Foundation of China (10171082)the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.
文摘We say a left R-module M is ecg-extending if every essentially countably generated submodule of M is essential in a direct summand of M. After giving some basic properties of ecg-extending modules, we show that for a nonsingular ring R, all left R-modules are ecg- extending if and only if all left R-modules are extending. We also characterize noetherian rings and artinian semisimple rings via ecg-quasi-continuous modules.
