Structural results on lifting, orthogonality and finiteness of idempotents View Full Text


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DATE

2021-12-27

AUTHORS

Abolfazl Tarizadeh, Pramod K. Sharma

ABSTRACT

In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring R is finite if and only if it is of the form 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{\kappa }$$\end{document} where κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is the cardinal of the connected components of Spec(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Spec}}(R)$$\end{document}. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature. More... »

PAGES

54

References to SciGraph publications

  • 1983-12. Lifting idempotents and Clifford theory in COMMENTARII MATHEMATICI HELVETICI
  • 2021-01-02. Structural results on harmonic rings and lessened rings in BEITRÄGE ZUR ALGEBRA UND GEOMETRIE / CONTRIBUTIONS TO ALGEBRA AND GEOMETRY
  • 2021-08-27. Structure theory of p.p. rings and their generalizations in REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES. SERIE A. MATEMÁTICAS
  • 2020-07-16. Characterizations of Gelfand Rings Specially Clean Rings and their Dual Rings in RESULTS IN MATHEMATICS
  • 1988-09. Lifting idempotents in near-rings in ARCHIV DER MATHEMATIK
  • 2021-09-30. Some Results on Pure Ideals and Trace Ideals of Projective Modules in ACTA MATHEMATICA VIETNAMICA
  • 1978. General Lattice Theory in NONE
  • 1992. Rings and Categories of Modules in NONE
  • 1991. A First Course in Noncommutative Rings in NONE
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    http://scigraph.springernature.com/pub.10.1007/s13398-021-01199-w

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    http://dx.doi.org/10.1007/s13398-021-01199-w

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