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A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

Part of: Representation theory of rings and algebras Abelian categories

Published online by Cambridge University Press:  07 October 2020

O. MENDOZA ,
M. ORTÍZ ,
C. SÁENZ  and
V. SANTIAGO
O. MENDOZA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mail: omendoza@matem.unam.mx
M. ORTÍZ
Affiliation:
Facultad de Ciencias, Universidad Autónoma del Estado de México, Mexico City, Mexico, e-mail: mortizmo@uaemex.mx
C. SÁENZ
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mails: corina.saenz@gmail.com, valente.santiago.v@gmail.com
V. SANTIAGO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico City, D.F. Mexico, e-mails: corina.saenz@gmail.com, valente.santiago.v@gmail.com
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Abstract

We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

MSC classification

Primary: 18E10: Exact categories, abelian categories
Secondary: 16G99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 1 - 36
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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