The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with $R$.Comment: A Remark was added at the end of the paper. To appear in Comm. Algebr

Nagata Rings, Kronecker Function Rings and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book \cite{G}) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Pr\"{u}fer and P. Lorenzen from 1930's. In \cite{FL1} and \cite{FL2} the current authors investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain $D$. In this paper we extend that study and also generalize Kang's notion of a star Nagata ring \cite{Kang:1987} and \cite{Kang:1989} to the semistar setting. Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on $D$.Comment: 20 page

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operationsComment: "Multiplicative Ideal Theory in Commutative Algebra: A tribute to the work of Robert Gilmer", Jim Brewer, Sarah Glaz, William Heinzer, and Bruce Olberding Editors, Springer (to appear

Cancellation properties in ideal systems: A classification of e.a.b.\boldsymbol{e.a.b.} semistar operations

We give a classification of {\texttt{e.a.b.}} semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all {\texttt{e.a.b.}} semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of {\texttt{e.a.b.}} semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that {\texttt{a.b.}} is really a stronger condition than {\texttt{e.a.b.}

Is there Space for "Genuine Autonomy" for Tibetan Areas in the PRC's System of Nationalities Regional Autonomy?

This article considers whether room exists within the current system of nationalities regional autonomy (NRA) in China to accommodate Tibetan aspirations for "genuine autonomy" under the People's Republic of China (PRC) sovereignty. It examines the legal framework for NRA in China, as well as Chinese government policy and practice toward autonomous areas, in terms of the limitations and possibilities they imply for realizing Tibetan aspirations for autonomy, highlighting specific areas of concern, opportunities and constraints. It explores the development of political and legal approaches toward autonomy since the 1930s, the nature of the current framework and how recent legal and political developments interact with that framework. It looks at options for autonomy under the Chinese Constitution and national legislation, particularly the self-government of nationality (minority) autonomous areas as well as Article 31 of the Constitution which has provided the basis for the establishment of special administrative regions (SARs). Since autonomous areas also exercise the general powers of local governments in the PRC, it describes the general system of local government at the provincial and lower administrative levels. The article examines the practical implementation and operation of minority autonomy and SARs including the apparent gap between law and practice. In particular, special attention is paid to the role of the Chinese Communist Party and its officials which have a significant impact on the exercise of state powers. It concludes that there are formidable obstacles to the autonomy that Tibetans seek in order to preserve their culture, values and identity. © Koninklijke Brill NV, Leiden, 2010.postprin