The State Space of K₀ of Exchange Rings

ABSTRACT

For a directly finite exchange ring R which satisfies general comparability, we construct all extreme points of the state space S(V(R),⟨ R⟩), where V(R) denotes the monoid of all isomorphic classes of finitely generated projective R-modules. From this, we further prove that S((K ₀(R),[R])) is affinely homeomorphic to M ₁ ⁺(BS(R)), where BS(R) denotes the spectrum of the Boolean algebra B(R) of all central idempotents in R, and M ⁺ ₁(BS(R)) the set of all probability measures on BS(R). These generalize the corresponding results on regular rings. Particularly, all of our results hold for exchange rings with all the idempotents central.

Key Words:

• Exchange rings
• General comparability
• Grothendieck groups
• State spaces

2000 Mathematics Subject Classification:

• 16E50
• 16E20

ACKNOWLEDGMENTS

The authors would like to thank the referee for excellent suggestions which helped us improve considerably the first version of the article. This work is supported by the National Natural Science Foundation of China (No. 10071035), the Natural Science Foundation of Hunan Province (No. 03JJY6017), and Hunan Province Education Committee Fund.

Notes

Communicated by B. Huisgen-Zimmermann.