Concerning P-frames and the Artin–Rees property

• Abedi, Mostafa ^[1]
  1. [1] Esfarayen University of Technology, Esfarayen, Iran
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 1, 2023, págs. 279-297
• Idioma: inglés
• DOI: 10.1007/s13348-021-00346-1
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let {\mathcal {R}}L be the ring of continuous real-valued functions on a completely regular frame L. The Artin–Rees property in {\mathcal {R}}L, in the factor rings of {\mathcal {R}}L and in the rings of fractions of {\mathcal {R}}L is studied. We show that a frame L is a P-frame if and only if {\mathcal {R}}L is an Artin–Rees ring if and only if every ideal of {\mathcal {R}}L with the Artin–Rees property is an Artin–Rees ideal if and only if the factor ring {\mathcal {R}}L/\langle \varphi \rangle is an Artin–Rees ring for any \varphi \in {\mathcal {R}}L. A necessary and sufficient condition for the local rings of a reduced ring to be Artin–Rees rings is that each of its prime ideals becomes minimal. It turns out that the local rings of {\mathcal {R}}L are an Artin–Rees ring if and only if L is a P-frame. We show that the complete ring of fractions of {\mathcal {R}}L is an Artin–Rees ring if and only if L is a cozero-complemented frame, or equivalently, the set of all minimal prime ideals of the ring {\mathcal {R}}L is compact. Finally, we prove that if \varphi \in {\mathcal {R}}L such that the open quotient \downarrow \!\!{{\,\mathrm{coz}\,}}\varphi is a dense C-quotient of L, then the ring of fractions ({\mathcal {R}}L)_\varphi is regular if and only if \downarrow \!\!{{\,\mathrm{coz}\,}}\varphi is a P-frame.

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