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 $S(K_0\pi).$

Keywords

Main Subjects


R. Alfaro, States on skew group rings and fixed rings, Comm. Algebra 18 (1990), no. 10, 3381--3394.

R. Alfaro, State spaces, finite algebras, and skew gorup rings, J. Algebra 139 (1991), no. 1, 134--154.

P. Ara, Extensions of exchange rings, J. Algebra 197 (1997), no. 2, 409--423.

B. Blackadar, K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ. 5, Springer-Verlag, New York, 1986.

B. Blackadar and M. Rørdam, Extending states on preordered semigroups and the existence of quasitraces on C*-algebras, J. Algebra 152 (1992), no. 1, 240--247.

P.M. Cohn, Free Rings and Their Relations, London Math. Society Monogr. Ser. 19, Academic Press, 2nd edition, London, 1985.

A. Facchini and D. Herbra, K0 of a semilocal ring, J. Algebra 225 (2000), no. 1, 47--69.

K.R. Goodearl, Partially Ordered Grothendieck Groups, Lecture Notes in Pure and Appl. Math. 91, Dekker, New York, 1984.

K.R. Goodearl and R.B. Warfield, State spaces of K0 of Noetherian rings, J. Algebra 71 (1981), no. 2, 322--378.

T.Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer-Verlag, New York, 1991.

B.A. Magurn, An Algebraic Introduction to K-theory, Encyclopedia Math. Appl. 87, Cambridge Univ. Press, Cambridge, 2002

J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, New York, 1987.

W.K. Nicholson and E. Sanchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra 271 (2004), no. 1, 391--406.

J. Rosenberg, Algebraic K-Theory and Its Applications, Springer-Verlag, New York, 1994.

J.R. Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, London-New York, 1981.

X. Zhu, Power stably free resolutions and Grothendieck groups, Comm. Algebra 29 (2001), no. 7, 2899--2921.

X. Zhu, Ranks of low K-groups of fibre products, Comm. Algebra 35 (2007), no. 1, 339--354.

X. Zhu, Lifting central idempotents modulo Jacobson radicals and ranks of K0 groups, Comm. Algebra 36 (2008), no. 8, 2833--2848.

X. Zhu, Affine maps of state spaces and state spaces of K0 groups, Math. Slovaca 63 (2013), no. 6, 1209--1226.

X. Zhu, Modules with local finite BIB-ranks and Grothendieck groups of some categories, Quaest. Math. 37 (2014), no. 1, 91--109.

X. Zhu and W. Tong, Categories of power stably free modules and their K0 groups, Sci. China Ser. A 40 (1997), no. 12, 1239--1246.


Volume 43, Issue 7
November and December 2017
Pages 2507-2516
  • Receive Date: 18 April 2017
  • Revise Date: 05 December 2017
  • Accept Date: 07 December 2017