On the maximal regular ideal of a linearly compact ring

1979 ◽  
Vol 33 (1) ◽  
pp. 232-234 ◽  
Author(s):  
Dinh Van Huynh
Keyword(s):  
Compact Ring ◽  
Regular Ideal ◽  
Maximal Regular Ideal

A CLOSURE OPERATION IN RINGS

2001 ◽  
Vol 12 (07) ◽  
pp. 791-812
Author(s):  
PERE ARA ◽  
GERT K. PEDERSEN ◽  
FRANCESC PERERA
Keyword(s):  
Regular Element ◽  
Closure Operation ◽  
Von Neumann ◽  
Regular Elements ◽  
C Algebras ◽  
Partial Inverse ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Invertible Elements ◽  
Maximal Regular Ideal

We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.

scholarly journals The maximal regular ideal of a ring

1950 ◽  
Vol 1 (2) ◽  
pp. 165-165 ◽  
Author(s):  
Bailey Brown ◽  
Neal H. McCoy
Keyword(s):  
Regular Ideal ◽  
Maximal Regular Ideal

On the maximal regular ideal of pointfree function rings, and more

2020 ◽  
Vol 273 ◽  
pp. 106960 ◽  
Author(s):  
Themba Dube
Keyword(s):  
Regular Ideal ◽  
Maximal Regular Ideal

The maximal regular ideal of self-injective and continuous rings splits off

1985 ◽  
Vol 44 (6) ◽  
pp. 511-521 ◽  
Author(s):  
Carl Faith
Keyword(s):  
Regular Ideal ◽  
Maximal Regular Ideal

scholarly journals On regular ideals in reduced rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3715-3726 ◽  
Author(s):  
A.R. Aliabady ◽  
R. Mohamadian ◽  
S. Nazari
Keyword(s):  
Closed Subset ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Closed Set ◽  
Von Neumann ◽  
Arbitrary Ring ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Isolated Points ◽  
Maximal Regular Ideal

Let R be a commutative ring with identity and X be a Tychonoff space. An ideal I of R is Von Neumann regular (briefly, regular) if for every a ? I, there exists b ? R such that a = a2b. In the present paper, we obtain the general form of a regular ideal in C(X) which is OA, for some closed subset A of ?X, for which Ac?X ? (P(X))?, where P(X) is the set of all P-points of X. We show that the ideals and subrings such as CK(X), C?(X), C?(X), SocmC(X) and M?X\X are regular if and only if they are equal to the socle of C(X). We carry further the study of the maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them OPD-spaces) the maximal regular ideal is OX\I(X), where I(X) is the set of isolated points of X. Also, for this class, the socle of C(X) is the maximal regular ideal if and only if I(X) contains no infinite closed set. We also show that C(X) contains an ideal which is both essential and regular if and only if (P(X))? is dense in X. Finally it is shown that, for semiprimitive rings pure ideals are of the form OA which A is a closed subset of Max(R), also a P-point of X = Max(R) is introduced and it is shown that the maximal regular ideal of an arbitrary ring R is OX\P(X), which P(X) is the set of P-points of X = Max(R).

scholarly journals The maximal regular ideal of the semigroup of binary relations

1981 ◽  
Vol 31 (2) ◽  
pp. 194-198
Author(s):  
D. Hardy ◽  
Francis J. Pastijn
Keyword(s):  
Binary Relations ◽  
Regular Ideal ◽  
Maximal Regular Ideal

Compact ring resonators using negative-refractive-index microstrip line

2005 ◽  
Vol 45 (4) ◽  
pp. 294-295 ◽  
Author(s):  
Aaron D. Scher ◽  
Christopher T. Rodenbeck ◽  
Kai Chang
Keyword(s):  
Refractive Index ◽  
Microstrip Line ◽  
Negative Refractive Index ◽  
Ring Resonators ◽  
Compact Ring

scholarly journals Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
E. Molnár ◽  
I. Prok ◽  
J. Szirmai
Keyword(s):  
Acta Math ◽  
Hyperbolic Manifolds ◽  
Space Forms ◽  
Hyperbolic Structures ◽  
Princeton University ◽  
Regular Ideal ◽  
Graphic Analysis ◽  
Novi Sad ◽  
Memorial Volume ◽  
János Bolyai

AbstractIn connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold $$M_{c^2}$$ M c 2 with double simplex $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 , seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of $$M_{c^2}={\widetilde{{\mathcal {D}}}}_1$$ M c 2 = D ~ 1 , leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 . Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on $$M_{c^2}$$ M c 2 as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

scholarly journals Air-trench splitters for ultra-compact ring resonators in low refractive index contrast waveguides

2008 ◽  
Vol 16 (1) ◽  
pp. 456 ◽  
Author(s):  
Nazli Rahmanian ◽  
Seunghyun Kim ◽  
Yongbin Lin ◽  
Gregory P. Nordin
Keyword(s):  
Refractive Index ◽  
Ring Resonators ◽  
Compact Ring ◽  
Refractive Index Contrast ◽  
Index Contrast ◽  
Low Refractive Index

scholarly journals The circumstellar environment of the B[e] star GG Car: an interferometric modeling

2014 ◽  
Vol 9 (S307) ◽  
pp. 291-292
Author(s):  
A. Domiciano de Souza ◽  
M. Borges Fernandes ◽  
A. C. Carciofi ◽  
O. Chesneau
Keyword(s):  
Binary System ◽  
Primary Component ◽  
Compact Ring ◽  
Hot Stars ◽  
Self Consistent ◽  
Grain Formation ◽  
Formation And Evolution ◽  
Circumstellar Environments ◽  
Consistent Treatment ◽  
Circumstellar Environment

AbstractThe research of stars with the B[e] phenomenon is still in its infancy, with several unanswered questions. Physically realistic models that treat the formation and evolution of their complex circumstellar environments are rare. The code HDUST (developed by A. C. Carciofi and J. Bjorkman) is one of the few existing codes that provides a self-consistent treatment of the radiative transfer in a gaseous and dusty circumstellar environment seen around B[e] supergiant stars. In this work we used the HDUST code to study the circumstellar medium of the binary system GG Car, where the primary component is probably an evolved B[e] supergiant. This system also presents a disk (probably circumbinary), which is responsible for the molecular and dusty signatures seen in GG Car spectra. We obtained VLTI/MIDI data on GG~Car at eight baselines, which allowed to spatially resolve the gaseous and dusty circumstellar environment. From the interferometric visibilities and SED modeling with HDUST, we confirm the presence of a compact ring, where the hot dust lies. We also show that large grains can reproduce the lack of structure in the SED and visibilities across the silicate band. We conclude the dust condensation site is much closer to the star than previously thought. This result provides stringent constraints on future theories of grain formation and growth around hot stars.

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