Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Modules with Non-Cyclic Socle and the Extension Property   In 2009, J. Wood proved that Frobenius bimodules have the extension property
for symmetrized weight compositions. More generally, it was later shown that
having a cyclic socle is sufficient for satisfying the property, while the
necessity remained an open question. In this thesis, a partial converse is
proved. For a significant class of finite module alphabets, the cyclic socle
condition is shown necessary for satisfying the extension property. The idea is
to use a new weight function to return to the original case of Hamming weight.

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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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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: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin