Spectrum of a Noncommutative Ring |
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| Authors: | Guoyin Zhang Wenting Tong Fanggui Wang |
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| Affiliation: | 1. Department of Basic Science , Jinling Institute of Technology , Nanjing, China;2. Department of Mathematics , Nanjing University , Nanjing, China gyzhang@jit.edu.cn gyzhangnju@sohu.com;4. Department of Mathematics , Nanjing University , Nanjing, China;5. Department of Mathematics , Sichuan Normal University , Chengdu, China |
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| Abstract: | R is any ring with identity. Let Spec r (R) (resp. Spec(R)) be the set of all prime right ideals (resp. all prime ideals) of R and let U r (eR) = {P ? Spec r (R) | e ? P}. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). A ring R is called Abelian if all idempotents in R are central (see Goodearl, 1991 Goodearl , K. R. ( 1991 ). Von Neumann Regular Rings. , 2nd ed. Malabar , Florida : Krieger Publishing Company . [Google Scholar]). A ring R is called 2-primal if every nilpotent element is in the prime radical of R (see Lam, 2001 Lam , T. Y. ( 2001 ). A First Course in Noncommutative Rings. , 2nd ed. (GTM 131) . New York : Springer-Verlag .[Crossref] , [Google Scholar]). It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen (i.e., closed and open) sets in Spec r (R). And the following results are obtained for any ring R: (1) For any clopen set U in Spec r (R), there is an idempotent e in R such that U = U r (eR). (2) If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r (eR) is a clopen set in Spec r (R). (3) Spec r (R) is connected if and only if Spec(R) is connected. |
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| Keywords: | Clopen idempotent Clopen set Connected space Prime 1-sided spectrum Top ring Weak Zariski topology |
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