KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2019 / Volume: 48 Issue: 6

First steps going down on algebraic frames

Themba Dube(University of South Africa)0000-0002-2702-2192
Sayfa
—
Yayın Tarihi
2019-12-08
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 $P$.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez:We extend the ring-theoretic concept of going down to algebraic frames and coherent maps.
📊
Temel Bulgular & Veri:We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1.
💡
Sonuç & Katkı: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 $P$.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Algebraic frame · Sublocale · Prime element

Abstract

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 $P$.

🔖

Bu Makaleye Atıf Yap

Dube, T. (2019). First steps going down on algebraic frames. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://doi.org/10.15672/HJMS.2018.638

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (17)

  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.