Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: K\"othe's Problem, Kurosch-Levitzki Problem and Graded Rings   Let $\mathfrak{R}$ be an associative ring graded by left cancellative monoid
$\mathsf{S}$, and $e$ the neutral element of $\mathsf{S}$. We study the
following problem: if $\mathfrak{R}_e$ is nil, then is $\mathfrak{R}$
nil/nilpotent? We have proved that if $\mathfrak{R}_e$ is nil (of bounded
index) and $\mathsf{f}$- commutative, then $\mathfrak{R}$ is nil (of bounded
index). Later, we have shown that $\mathfrak{R}_e$ being nilpotent implies
$\mathfrak{R}$ is nilpotent. Consequently, we have exhibited a generalization
of Dubnov-Ivanov-Nagata-Higman Theorem for the graded algebras case.
Furthermore, we have exhibited relations between graded rings and the problems
of K\"{o}the and Kurosh-Levitzki. We have proved that graded rings and
$\mathsf{f}$-commutative rings provide positive solutions to these problems.

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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
1
The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph