A coordinatization functor L(−,1):Ring→Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R↦L(R) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective (−)op endofunctors on Ring and Latt; and 3) it admits localization at a left R-module. The complemented elements of L(R,1) form a partially ordered set S(R) isomorphic to the space of direct summands of RR. The right nonsingular rings for which the embedding of S(R) into the localization L(R,1)Q at the right maximal ring of quotients QR is an isomorphism are characterized by a property that every finite matrix subgroup φ(RR) of the left R-module RR is essential in an element of S(R). In that case, the space S(R) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

The lattice coordinatized by a ring

L'Innocente S.
Co-primo
2023-01-01

Abstract

A coordinatization functor L(−,1):Ring→Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R↦L(R) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective (−)op endofunctors on Ring and Latt; and 3) it admits localization at a left R-module. The complemented elements of L(R,1) form a partially ordered set S(R) isomorphic to the space of direct summands of RR. The right nonsingular rings for which the embedding of S(R) into the localization L(R,1)Q at the right maximal ring of quotients QR is an isomorphism are characterized by a property that every finite matrix subgroup φ(RR) of the left R-module RR is essential in an element of S(R). In that case, the space S(R) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.
2023
262
File in questo prodotto:
File Dimensione Formato  
1-s2.0-S0022404922002080-main.pdf

solo gestori di archivio

Tipologia: Versione Editoriale
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 514.63 kB
Formato Adobe PDF
514.63 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11581/466092
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact