# State spaces of $K_0$ groups of some rings

Document Type: Research Paper

Author

Audio-visual Center‎, ‎The Nanjing Institute of Tourism and Hospitality‎, ‎211100‎, ‎Nanjing‎, ‎P.R‎. ‎China.

Abstract

‎Let $R$ be a ring‎ ‎with the Jacobson radical $J(R)$ and let $\pi\colon R\to R/J(R)$ be‎ the canonical map‎. ‎Then $\pi$ induces an order preserving group homomorphism‎ ‎$K_0\pi\colon K_0(R)\to K_0(R/J(R))$ and an‎ ‎affine continuous map $S(K_0\pi)$ between the state space $St(R/J(R))$ and the‎ ‎state space $St(R).$‎ ‎In this paper‎, ‎we consider the natural affine map $S(K_0\pi).$ We give a condition under which $S(K_0\pi)$ is‎ ‎an affine homeomorphism‎. ‎At the same time‎, ‎we discuss the relationship between semilocal rings and semiperfect rings by using the‎ ‎affine map $S(K_0\pi).$

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### History

• Receive Date: 18 April 2017
• Revise Date: 05 December 2017
• Accept Date: 07 December 2017