The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings

We investigate the prime spectrum of a noncommutative ring and its spectral closure, the extended prime spectrum. We construct a ring for which the prime spectrum is a spectral space different from the extended prime spectrum and we construct a von Neumann regular ring for which the prime spectrum is not a spectral space. The authors are members of the European Research Training Network RAAG (Contract No. HPRN-CT-2001-00271). The first author was also supported by the Ministry of Education, Science and Sport of Slovenia.