Sunsook Noh

Simple complete ideals in two-dimensional regular local rings

In a two-dimensional regular local ring (R,m), it is known that there exists a unique complete ideal I adjacent to a given simple complete m-primary ideal J from above. In this paper it is shown that there are infinitely many simple complete m-primary ideals adjacent to a given simple complete m-primary ideal J (≠ m) from below whose orders are the same as that of J and that there exists a unique complete m-primary ideal adjacent to J from below whose order is one bigger than that of J. We also show that these are all the complete ideals adjacent to J from below. It is known that there is a unique prime divisor w and a unique infinitely near point S of R associated to a given simple complete m-primary ideal J. As a corollary of the main theorem, we obtain one-to-one correspondences between the set of simple m-primary complete ideals adjacent to J from below, the set of first neighborhood prime divisors of w and the set of first quadratic transformations of S.

An e-learning framework supporting personalization and collaboration

Advances in Internet technology have made possible, rapid adoption of e-learning in many fields of society. Various educational methods for e-learning have been developed because personalization can be easily implemented through web services. However, these e-learning methods are not widely used because it is difficult for instructors to apply them in developing their courseware. Currently, the most widely used style of e-learning courseware is viewing video clips of lectures and discussing the lecture using bulletin boards. It is an electronic version of the traditional classroom. In this paper, we propose a user-friendly e-learning framework that supports personalization and collaboration between the instructor and the students. Instructors can create various levels of learning objects and define the relationships between the objects. Students can select the topics of their interest and dynamically adjust the level of learning in real time. Multiple instructors can cooperate in expanding the courseware. All the participants can also annotate the learning materials independently.

Internet at a crossroads of education

The Internet has grown rapidly in the past few years to impact all aspects of modern life. While many areas of the Internet such as commerce has shown tremendous improvements in providing convenient and complex transaction services from shopping to banking, the area of education has seen less development in terms of meeting the potential of the Internet for education. The number of web sites related to education and the types of educational uses have clearly increased on the Internet but the overall consensus is that more should be done in the area of using the Internet to enhance the educational experience of learners. Many technological innovations have been previously touted as a new paradigm for education only to see it disappear from view quietly with time. In order to make sure that the vast potential of the Internet is fully utilized for education, more structured approach and a change in the mindset of educators are needed.

Educational use of E-mentoring to encourage women into science and engineering

An important emerging area of Internet use for education is electronic mentoring or e-mentoring. The Internet provides an ideal communication medium that is both broad and flexible enough to provide unique opportunities for students and mentors. Today, women workers represent a minority in the fields of science and engineering. In Korea, a government sponsored pilot program Women into Science and Engineering (WISE) was begun in 2001 at Ewha Womans University to encourage female students at all school levels to pursue careers in science and engineering. Within the WISE program, an e-mentoring pilot study based on a bulletin board system was developed and tested as an educational tool for encouraging female students. The following conclusions were determined from the pilot study. While the survey of students indicated a positive response to the e-mentoring program, analysis of the bulletin board data revealed that less than half of the students actively participated in the program. The study suggests that e-mentoring programs require more than just a technical and administrative framework to be a success. It requires an active and constant feedback system to ensure that everyone is participating in the program.

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.

SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by . In this paper, if the m-adic order of P is 2, we show that . We also show that when w is the prime divisor associated to a simple v-ideal of order 2 and that w(R) = v(R) as well.

A Vision for Improving Mathematics Education in the Internet-Based Society

The development of the Internet has brought new opportunities for education. As more and more education related web sites appear daily,there is a need to determine the framework with which the Internet could be best used for education. One vision for the framework based on experiences with mathematics education is the following. First, use the Internet to help teachers to become better teachers. Second, develop web-based instruction contents that are coordinated with the curriculum and the textbook for teachers to use in the classroom. Finally, support more research to determine the best ways of developing web-based contents that show clear improvement in learning. The vision is based on the assumption that teachers,esp ecially in the K-12 grade levels,will continue to be the most important part of the educational process for students even in the Internet-based society.

SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS

Let (R, m) be a 2-dimensional regular local ring with alge- braically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals m = P0 ⊃ P1 ⊃ · · · ⊃ Pt = P and all the other v-ideals are uniquely factored into a product of those simple ones (17). Lipman further showed that the predecessor of the smallest simple v-ideal P is either simple or the product of two simple v-ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple v-ideals when P is satellite of order 3 in terms of the invariant bv = |v(x) − v(y)|, where v is the prime divisor associated to P and m = (x, y). Denote bv by b and let b = 3k + 1 for k = 0, 1, 2. Let ni be the number of nonmaximal simple v-ideals of order i for i = 1, 2, 3. We show that the numbers nv = (n1, n2, n3) = (⌈ b+1 3 ⌉, 1, 1) and that the rank of P is ⌈ b+7 3 ⌉ = k + 3. We then describe all the v-ideals from m to P as products of those simple v-ideals. In particular, we find the conductor ideal and the v-predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for k ≥ 1. We also find the value semigroup v(R) of a satellite simple valuation ideal P of order 3 in terms of bv.

Integrally Closed Modules and Their Divisors

ABSTRACT There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ ideal obtained through Noether normalization. They are useful to describe the integral closure of some class of modules and to study the completeness of the modules of Kähler differentials.