Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $F$-nilpotent rings and permanence properties   We explore the singularity classes $F$-nilpotent, weakly $F$-nilpotent, and
generalized weakly $F$-nilpotent under faithfully flat local ring maps. As an
application, we show that the loci of primes in a Noetherian ring of prime
characteristic which define either weakly $F$-nilpotent or $F$-nilpotent local
rings are open with respect to the Zariski topology whenever $R$ is $F$-finite
or essentially of finite type over an excellent local ring.

18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
10
The categories of this paper are: q-bio