IBN Rings and Orderings on Grothendieck Groups  

作　　者：Tong Wenting Department of Mathematics Nanjing University Nanjing,210008 China 

出　　处：《Acta Mathematica Sinica,English Series》1994年第3期225-230,共6页数学学报（英文版）

基　　金：Supported by National Nature Science Foundation of China.

摘　　要：Let R be a ring with an identity element.R∈IBN means that R^m■Rⁿ implies m=n,R ∈IBN₁ means that R^m ■Rⁿ⊕K implies m≥n,and R ∈IBN₂ means that R^m■R^m⊕K implies K=0.In this paper we give some characteristic properties of IBN₁ and IBN₂,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM₁ and all finitely generated projective left R-modules are stably free,then the Grothendieck group K_o（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K_o（R）of a ring R is a partial ordering,then R ∈IBM₁ or K_o（R）=0.Let R be a ring with an identity element.R∈IBN means that R^m■Rⁿ implies m=n,R ∈IBN₁ means that R^m ■Rⁿ⊕K implies m≥n,and R ∈IBN₂ means that R^m■R^m⊕K implies K=0.In this paper we give some characteristic properties of IBN₁ and IBN₂,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM₁ and all finitely generated projective left R-modules are stably free,then the Grothendieck group K_o（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group K_o（R）of a ring R is a partial ordering,then R ∈IBM₁ or K_o（R）=0.

关 键 词：IBN Rings and Orderings on Grothendieck Groups MATH PSF 

分 类 号：O1[理学—数学]