Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The small finitistic dimensions over commutative rings   Let $R$ be a commutative ring with identity. The small finitistic dimension
$\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of
$R$-modules with finite projective resolutions. In this paper, we characterize
a ring $R$ with $\fPD(R)\leq n$ using finitely generated semi-regular ideals,
tilting modules, cotilting modules of cofinite type or vaguely associated prime
ideals. As an application, we obtain that if $R$ is a Noetherian ring, then
$\fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\}$ where $\grade(\m,R)$ is the grade
of $\m$ on $R$ . We also show that a ring $R$ satisfies $\fPD(R)\leq 1$ if and
only if $R$ is a $\DW$ ring. As applications, we show that the small finitistic
dimensions of strong \Prufer\ rings and $\LPVD$s are at most one. Moreover, for
any given $n\in \mathbb{N}$, we obtain examples of total rings of quotients $R$
with $\fPD(R)=n$.

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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas