FAITH'S PROBLEM ON R-PROJECTIVITY IS UNDECIDABLE
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10401396" target="_blank" >RIV/00216208:11320/19:10401396 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=FynXWlgi0b" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=FynXWlgi0b</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/proc/14209" target="_blank" >10.1090/proc/14209</a>
Alternative languages
Result language
angličtina
Original language name
FAITH'S PROBLEM ON R-PROJECTIVITY IS UNDECIDABLE
Original language description
In Faith [Grundlehren der Mathematischen Wissenschaften. 191 (1976)], Faith asked for what rings R does the Dual Baer Criterion hold in Mod-R, that is, when does R-projectivity imply projectivity for all right R-modules? Such rings R were called right testing. Sandomierski proved that all right perfect rings are right testing. Puninski et al. [J. Algeb. 484 (2017) pp. 198-206] have recently shown for a number of nonright perfect rings that they are not right testing, and noticed that [Trans. Amer. Math. Soc. 348 (1996) pp. 1521-1554] proved consistency with ZFC of the statement 'each right testing ring is right perfect' (the proof used Shelah's uniformization). Here, we prove the complementing consistency result: the existence of a right testing, but not right perfect ring is also consistent with ZFC (our proof uses Jensen-functions). Thus the answer to the Faith's question above is undecidable in ZFC. We also provide examples of nonright perfect rings such that the Dual Baer Criterion holds for all small modules (where small means countably generated, or <= 2(N0)-presented of projective dimension <= 1).
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA17-23112S" target="_blank" >GA17-23112S: Structure theory for representations of algebras (localization and tilting theory)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Proceedings of the American Mathematical Society
ISSN
0002-9939
e-ISSN
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Volume of the periodical
147
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
8
Pages from-to
497-504
UT code for WoS article
000454742000008
EID of the result in the Scopus database
2-s2.0-85061568935