First steps going down on algebraic frames
Themba DUBE
^{
[1]
}
We extend the ringtheoretic concept of going down to algebraic frames and coherent maps. We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1. An application to rings yields a characterization of von Neumann regular rings that appears to have hitherto been overlooked. Namely, a commutative ring $A$ with identity is von Neumann regular if and only if $Ann(I)+P=A$, for every prime ideal $P$ of $A$ and any finitely generated ideal $I$ of $A$ contained in $P$.
Algebraic frame, Sublocale, Prime element, Goingdown, Commutative ring, Dimension
 [1] C.E. Aull and W.J. Thron, Separation axioms between $T_0$ and $T_1$, Indag. Math. 24,
26–37, 1962.
 [2] B. Banaschewski, Radical ideals and coherent frames, Comment. Math. Univ. Carolin.
37, 349–370, 1996.
 [3] B. Banaschewski, Gelfand and exchange rings: their spectra in pointfree topology,
Arab. J. Science and Engineering 25, 3–22, 2003.
 [4] B. Banaschewski and A. Pultr, Variants of openness, Appl. Categ. Structures 2,
331–350, 1994.
 [5] B. Banaschewski and A. Pultr, Pointfree aspects of the $T_D$ axiom of classical topology,
Quaest. Math. 33, 369–385, 2010.
 [6] T. Coquand and H. Lombardi, Hidden constructions in abstract algebra: Krull dimension
of distributive lattices and commutative rings, in: Commutative ring theory
and applications 477–499, Fez, 2001, Lecture Notes in Pure and Appl. Math., 231,
Dekker, New York, 2003.
 [7] D.E. Dobbs and M. Fontana, Classes of commutative rings characterized by GoingUp
and GoingDown behavior, Rend. Sem. Mat. Univ. Padova 66, 113–127, 1982.
 [8] D.E. Dobbs and I.J. Papick, Going down: a survey, Nieuw Arch. Wisk. 26, 255–291,
1978.
 [9] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc.
142, 43–60, 1969.
 [10] P.T. Johnstone, Stone Spaces. Cambridge University Press, Cambridge, 1982.
 [11] J. Martínez, Archimedean lattices, Algebra Universalis 3, 247–260, 1973.
 [12] J. Martínez, Dimension in algebraic frames, Czechoslovak Math. J. 56, 437–474, 2006.
 [13] J. Martínez, Unit and kernel systems in algebraic frames, Algebra Universalis 55,
13–43, 2006.
 [14] J. Martínez, An innocent theorem of Banaschewski, applied to an unsuspecting theorem
of De Marco, and the aftermath thereof, Forum Math. 23, 565–596, 2013.
 [15] J. Martínez and E.R. Zenk, When an algebraic frame is regular, Algebra Universals
50, 231–257, 2003.
 [16] S.B. Niefield and K.I. Rosenthal, Componental nuclei, in: Categorical algebra and
its applications (LouvainLaNeuve, 1987), 299–306, Lecture Notes in Math., 1348,
Springer, Berlin, 1988.
 [17] J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in
Mathematics, Springer, Basel, 2012.
Primary Language 
en

Subjects 
Mathematics

Journal Section 
Mathematics 
Authors 
Orcid: 0000000227022192 Author: Themba DUBE (Primary Author) Institution: University of South Africa Country: South Africa

Dates 
Publication Date
: December 8, 2019

Bibtex 
@research article { hujms480650,
journal = {Hacettepe Journal of Mathematics and Statistics},
issn = {2651477X},
eissn = {2651477X},
address = {},
publisher = {Hacettepe University},
year = {2019},
volume = {48},
pages = {1792  1807},
doi = {10.15672/HJMS.2018.638},
title = {First steps going down on algebraic frames},
key = {cite},
author = {DUBE, Themba}
} 
APA

DUBE, T
.
(2019).
First steps going down on algebraic frames.
Hacettepe Journal of Mathematics and Statistics
, 48 (6) ,
17921807 .
DOI: 10.15672/HJMS.2018.638 
MLA

DUBE, T
.
"First steps going down on algebraic frames".
Hacettepe Journal of Mathematics and Statistics 48 (2019
): 17921807 <https://dergipark.org.tr/en/pub/hujms/issue/50516/480650>

Chicago

DUBE, T
.
"First steps going down on algebraic frames".
Hacettepe Journal of Mathematics and Statistics 48 (2019
): 17921807 
RIS 
TY  JOUR
T1  First steps going down on algebraic frames
AU  Themba DUBE
Y1  2019
PY  2019
N1
 doi: 10.15672/HJMS.2018.638 DO
 10.15672/HJMS.2018.638 T2  Hacettepe Journal of Mathematics and Statistics
JF  Journal
JO  JOR
SP  1792
EP  1807
VL  48
IS  6
SN  2651477X2651477X
M3
 doi: 10.15672/HJMS.2018.638 UR
 https://doi.org/10.15672/HJMS.2018.638 Y2  2018
ER 

EndNote 
%0 Hacettepe Journal of Mathematics and Statistics First steps going down on algebraic frames
%A Themba DUBE
%T First steps going down on algebraic frames
%D 2019
%J Hacettepe Journal of Mathematics and Statistics
%P 2651477X2651477X
%V 48
%N 6
%R doi: 10.15672/HJMS.2018.638 %U 10.15672/HJMS.2018.638 
ISNAD 
DUBE, Themba
.
"First steps going down on algebraic frames". Hacettepe Journal of Mathematics and Statistics
48
/
6
(December 2019):
17921807
. https://doi.org/10.15672/HJMS.2018.638 
AMA 
DUBE T
.
First steps going down on algebraic frames.
Hacettepe Journal of Mathematics and Statistics.
2019;
48(6):
17921807.

Vancouver 
DUBE T
.
First steps going down on algebraic frames.
Hacettepe Journal of Mathematics and Statistics.
2019;
48(6):
18071792.
