Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Faith's problem on R-projectivity is undecidable   In \cite{F}, 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 if $R$ is right perfect, then $R$ is right testing. Puninski et
al.\ \cite{AIPY} have recently shown for a number of non-right perfect rings
that they are not right testing, and noticed that \cite{T2} proved consistency
with ZFC of the statement {\lq}each right testing ring is right perfect{\rq}
(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 non-right perfect rings such that the Dual
Baer Criterion holds for {\lq}small{\rq} modules (where {\lq}small{\rq} means
countably generated, or $\leq 2^{\aleph_0}$-presented of projective dimension
$\leq 1$).

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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: dg-ga