Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of artinian partially ordered sets over semiartinian Von
  Neumann regular algebras   If $R$ is a semiartinian Von Neumann regular ring, then the set $\Prim_{R}$
of primitive ideals of $R$, ordered by inclusion, is an artinian poset in which
all maximal chains have a greatest element. Moreover, if $\Prim_{R}$ has no
infinite antichains, then the lattice $\BL_{2}(R)$ of all ideals of $R$ is
anti-isomorphic to the lattice of all upper subsets of $\Prim_{R}$. Since the
assignment $U\mapsto r_R(U)$ defines a bijection from any set $\Simp_R$ of
representatives of simple right $R$-modules to $\Prim_{R}$, a natural partial
order is induced in $\Simp_R$, under which the maximal elements are precisely
those simple right $R$-modules which are finite dimensional over the respective
endomorphism division rings; these are always $R$-injective. Given any artinian
poset $I$ with at least two elements and having a finite cofinal subset, a
lower subset $I'\sbs I$ and a field $D$, we present a construction which
produces a semiartinian and unit-regular $D$-algebra $D_I$ having the following
features: (a) $\Simp_{D_I}$ is order isomorphic to $I$; (b) the assignment
$H\mapsto\Simp_{D_I/H}$ realizes an anti-isomorphism from the lattice
$\BL_{2}(D_I)$ to the lattice of all upper subsets of $\Simp_{D_I}$; (c) a
non-maximal element of $\Simp_{D_I}$ is injective if and only if it corresponds
to an element of $I'$, thus $D_I$ is a right $V$-ring if and only if $I' = I$;
(d) $D_I$ is a right \emph{and} left $V$-ring if and only if $I$ is an
antichain; (e) if $I$ has finite dual Krull length, then $D_I$ is (right and
left) hereditary; (f) if $I$ is at most countable and $I' = \vu$, then $D_I$ is
a countably dimensional $D$-algebra.

31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
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The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph