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.
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Supported by National Nature Science Foundation of China.
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Wenting, T. IBN rings and orderings on grothendieck groups. Acta Mathematica Sinica 10, 225–230 (1994). https://doi.org/10.1007/BF02560713
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DOI: https://doi.org/10.1007/BF02560713