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.}

Heterozygous deletion of both sclerostin (Sost) and connexin43 (Gja1) genes in mice is not sufficient to impair cortical bone modeling

Connexin43 (Cx43) is the main gap junction protein expressed in bone forming cells, where it modulates peak bone mass acquisition and cortical modeling. Genetic ablation of the Cx43 gene (Gja1) results in cortical expansion with accentuated periosteal bone formation associated with decreased expression of the Wnt inhibitor sclerostin. To determine whether sclerostin (Sost) down-regulation might contribute to periosteal expansion in Gja1 deficient bones, we took a gene interaction approach and crossed mice harboring germline null alleles for Gja1 or Sost to generate single Gja1+/-and Sost+/-and double Gja1+/-;Sost+/-heterozygous mice. In vivo μCT analysis of cortical bone at age 1 and 3 months confirmed increased thickness in Sost-/-mice, but revealed no cortical abnormalities in single Gja1+/-or Sost+/-mice. Double heterozygous Gja1+/-Sost+/-also showed no differences in mineral density, cortical thickness, width or geometry relative to wild type control mice. Likewise, 3-point bending measurement of bone strength revealed no significant differences between double Gja1+/-;Sost+/-or single heterozygous and wild type mice. Although these data do not exclude a contribution of reduced sclerostin in the cortical expansion seen in Gja1 deficient bones, they are not consistent with a strong genetic interaction between Sost and Gja1 dictating cortical modeling

Nuclear electric propulsion development and qualification facilities

This paper summarizes the findings of a Tri-Agency panel consisting of members from the National Aeronautics and Space Administration (NASA), U.S. Department of Energy (DOE), and U.S. Department of Defense (DOD) that were charged with reviewing the status and availability of facilities to test components and subsystems for megawatt-class nuclear electric propulsion (NEP) systems. The facilities required to support development of NEP are available in NASA centers, DOE laboratories, and industry. However, several key facilities require significant and near-term modification in order to perform the testing required to meet a 2014 launch date. For the higher powered Mars cargo and piloted missions, the priority established for facility preparation is: (1) a thruster developmental testing facility, (2) a thruster lifetime testing facility, (3) a dynamic energy conversion development and demonstration facility, and (4) an advanced reactor testing facility (if required to demonstrate an advanced multiwatt power system). Facilities to support development of the power conditioning and heat rejection subsystems are available in industry, federal laboratories, and universities. In addition to the development facilities, a new preflight qualifications and acceptance testing facility will be required to support the deployment of NEP systems for precursor, cargo, or piloted Mars missions. Because the deployment strategy for NEP involves early demonstration missions, the demonstration of the SP-100 power system is needed by the early 2000's

Ultrafilter and Constructible topologies on spaces of valuation domains

Let $K$ be a field and let $A$ be a subring of $K$. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar$(K|A)$ of all valuation domains having $K$ as quotient field and containing $A$. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar$(K|A)$. We extend results regarding distinguished spectral topologies on spaces of valuation domains.Comment: Comm. Algebra (accepted for publication