Using Survival Pairs to Characterize Rings of Algebraic Integers

Dobbs, David Earl Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.2

Let R be a domain with quotient field K and prime subring A. Then R is integral over each of its subrings having quotient field K if and only if (A, R) is a survival pair. This shows the redundancy of a condition involving going-down pairs in a earlier characterization of such rings. In characteristic 0, the domains being characterized are the rings R that are isomorphic to subrings of the ring of all algebraic integers. In positive (prime) characteristic, the domains R being characterized are of two kinds: either R = K is an algebraic field extension of A or precisely one valuation domain of K does not contain R.

Some Analogues of a Result of Vasconcelos

DOBBS, DAVID EARL,SHAPIRO, JAY ALLEN Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.4

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (${\star}$) if for each nonzero element $a{\in}R$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (${\star}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (${\star}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (${\star}$), but the converse is false.

Normal Pairs of Going-down Rings

Dobbs, David Earl,Shapiro, Jay Allen Department of Mathematics 2011 Kyungpook mathematical journal Vol.51 No.1

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

On the Subsemigroups of a Finite Cyclic Semigroup

Dobbs, David Earl,Latham, Brett Kathleen Department of Mathematics 2014 Kyungpook mathematical journal Vol.54 No.4

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.

The Factor Domains that Result from Uppers to Prime Ideals in Polynomial Rings

Dobbs, David Earl Department of Mathematics 2010 Kyungpook mathematical journal Vol.50 No.1

Let P be a prime ideal of a commutative unital ring R; X an indeterminate; D := R/P; L the quotient field of D; F an algebraic closure of L; ${\alpha}$ ${\in}$ L[X] a monic irreducible polynomial; ${\xi}$ any root of in F; and Q = <P, ${\alpha}$>, the upper to P with respect to ${\alpha}$. Then R[X]/Q is R-algebra isomorphic to $D[{\xi}]$; and is R-isomorphic to an overring of D if and only if deg(${\alpha}$) = 1.

Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

Dobbs, David Earl Department of Mathematics 2020 Kyungpook mathematical journal Vol.60 No.1

Let (A, M) ⊂ (B, N) be commutative quasi-local rings. We consider the property that there exists a ring D such that A ⊆ D ⊂ B and the extension D ⊂ B is inert. Examples show that the number of such D may be any non-negative integer or infinite. The existence of such D does not imply M ⊆ N. Suppose henceforth that M ⊆ N. If the field extension A/M ⊆ B/N is algebraic, the existence of such D does not imply that B is integral over A (except when B has Krull dimension 0). If A/M ⊆ B/N is a minimal field extension, there exists a unique such D, necessarily given by D = A + N (but it need not be the case that N = MB). The converse fails, even if M = N and B/M is a finite field.

Thailand’s Kra Isthmus and Elusive Canal Plans since the 1850s

Stephen Dobbs 서강대학교 동아연구소 2016 TRaNS(Trans –Regional and –National Studies of Sou Vol.4 No.1

This article uses primarily British colonial documents and Singapore sources to examine the history ofproposals for the construction ofa shipping canal through Thailand’s Kra Isthmus from the 1850s. It provides historical background for continuing interest in constructing a Kra canal with the most recent speculative discussions involving Chinese interests. Since the 1850s the idea of a canal was revived on numerous occasions with several detailed surveys conducted over this time to assess the feasibility of a shipping canal via the Kra Isthmus. This research examines how speculation and actual proposals were handled by the British colonial authorities and how this related to the British policy of using Siam/Thailand as a buffer state separating their colonies and those of their European rival France. In colonial Singapore canal proposals created great angst and to some extent this has continued to be the case to the present day. The article suggests that while British colonial policy was always against, or at least not in favour of, the construction of a canal, other factors are equally important for explaining why canal proposals never proceeded beyond planning and surveys.

Engaging Communities in Area-based Regeneration : The Role of Participatory Evaluation

Lynn Dobbs,Craig Moore 행정언어와 질적연구학회 2010 행정언어와 질적연구 Vol.1 No.1

Slowly breaking waves: the longevity of tidally induced spiral structure

Struck, Curtis,Dobbs, Clare L.,Hwang, Jeong‐,Sun Blackwell Publishing Ltd 2011 MONTHLY NOTICES- ROYAL ASTRONOMICAL SOCIETY Vol.414 No.3

<P><B>ABSTRACT</B></P><P>We have discovered long‐lived waves in two sets of numerical models of fast (marginally bound or unbound) flyby galaxy collisions, carried out independently with two different codes. In neither simulation set are the spirals the result of a collision‐induced bar formation. Although there is variation in the appearance of the waves with time, they do not disappear and reform recurrently, as seen in other cases described in the literature. We also present an analytic theory that can account for the wave structure, not as propagating transients, nor as a fixed pattern propagating through the disc. While these waves propagate through the disc, they are mantained by the coherent oscillations initiated by the impulsive disturbance. Specifically, the analytic theory suggests that they are caustic waves in ensembles of stars pursuing correlated epicyclic orbits after the disturbance. This theory is an extension of that developed by Struck and collaborators for colliding ring galaxies.</P><P>The models suggest that this type of wave may persist for a couple of Gyr, and galaxy interactions occur on comparable time‐scales, so waves produced by the mechanism may be well represented in observed spirals. In particular, this mechanism can account for the tightly wound, and presumably long‐lived, spirals seen in some nearby early‐type galaxies. These spirals are also likely to be common in groups and clusters, where fast encounters between galaxies occur relatively frequently. However, as the spirals become tightly wound, and evolve to modest amplitudes, they may be difficult to resolve unless they are nearby. None the less, the effect may be one of several processes that result from galaxy harassment, and via wave‐enhanced star formation, contributes to the Butcher–Oemler effect.</P>

Vacuum Electronic High Power Terahertz Sources

Booske, J. H.,Dobbs, R. J.,Joye, C. D.,Kory, C. L.,Neil, G. R.,Gun-Sik Park,Jaehun Park,Temkin, R. J. IEEE 2011 IEEE transactions on terahertz science and technol Vol.1 No.1

<P>Recent research and development has been incredibly successful at advancing the capabilities for vacuum electronic device (VED) sources of powerful terahertz (THz) and near-THz coherent radiation, both CW or average and pulsed. Currently, the VED source portfolio covers over 12 orders of magnitude in power (mW-to-GW) and two orders of magnitude in frequency (from <; 0.1 to >; 10 THz). Further advances are still possible and anticipated. They will be enabled by improved understanding of fundamental beam-wave interactions, electromagnetic mode competition and mode control, along with research and development of new materials, fabrication methods, cathodes, electron beam alignment and focusing, magnet technologies, THz metrology and advanced, broadband output radiation coupling techniques.</P>