422 research outputs found
The space of finitely generated rings
The space of marked commutative rings on n given generators is a compact
metrizable space. We compute the Cantor-Bendixson rank of any member of this
space. For instance, the Cantor-Bendixson rank of the free commutative ring on
n generators is omega^n, where omega is the smallest infinite ordinal. More
generally, we work in the space of finitely generated modules over a given
commutative ring.Comment: 10 pages, no figure. To appear in Internat. J. Algebra Compu
Counting submodules of a module over a noetherian commutative ring
We count the number of submodules of an arbitrary module over a countable
noetherian commutative ring. We give, along the way, a structural description
of meager modules, which are defined as those that do not have the square of a
simple module as subquotient. We deduce in particular a characterization of
uniserial modules over commutative noetherian rings.Comment: 34 pages. v2: expanded introduction and preliminarie
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