IBN rings and orderings on grothendieck groups |
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Authors: | Tong Wenting |
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Institution: | (1) Department of Mathematics, Nanjing University, 210008 Nanjing, China |
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Abstract: | LetR be a ring with an identity element.R∈IBN means thatR
m⋟Rn impliesm=n, R∈IBN
1 means thatR
m⋟Rn⊕K impliesm≥n, andR∈IBN
2 means thatR
m⋟Rm⊕K impliesK=0. In this paper we give some characteristic properties ofIBN
1 andIBN
2, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN
1 and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK
0(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK
0(R) of a ringR is a partial ordering, thenR∈IBN
1 orK
0(R)=0.
Supported by National Nature Science Foundation of China. |
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Keywords: | |
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