Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modules Satisfying the Prime Radical Condition and a Sheaf Construction
  for Modules I   The purpose of this paper and its sequel, is to introduce a new class of
modules over a commutative ring $R$, called $\mathbb{P}$-radical modules
(modules $M$ satisfying the prime radical condition
"$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for every prime ideal
${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{{\cal{P}}M}$ is the
intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This
class contains the family of primeful modules properly. This yields that over
any ring all free modules and all finitely generated modules lie in the class
of $\mathbb{P}$-radical modules. Also, we show that if $R$ is a domain (or a
Noetherian ring), then all projective modules are $\mathbb{P}$-radical. In
particular, if $R$ is an Artinian ring, then all $R$-modules are
$\mathbb{P}$-radical and the converse is also true when $R$ is a Noetherian
ring. Also an $R$-module $M$ is called $\mathbb{M}$-radical if
$(\sqrt[p]{{\cal{M}}M}:M)={\cal{M}}$; for every maximal ideal
${\cal{M}}\supseteq {\rm Ann}(M)$. We show that the two concepts
$\mathbb{P}$-radical and $\mathbb{M}$-radical are equivalent for all
$R$-modules if and only if $R$ is a Hilbert ring. Semisimple
$\mathbb{P}$-radical ($\mathbb{M}$-radical) modules are also characterized. In
Part II we shall continue the study of this construction, and as an
application, we show that the sheaf theory of spectrum of $\mathbb{P}$-radical
modules (with the Zariski topology) resembles to that of rings.

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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: q-alg