First steps going down on algebraic frames
We extend the ring-theoretic 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 $P$.
Keywords:
Algebraic frame, Sublocale, Prime element, Going-down, 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 Going-Up and Going-Down 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 (Louvain-La-Neuve, 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.
- Yayın Aralığı: Yılda 6 Sayı
- Başlangıç: 2002
- Yayıncı: Hacettepe Üniversitesi Fen Fakultesi