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finitely generated PRID.

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Every finitely generated abelian group or nilpotent group is polycyclic.
en.wikipedia.org
One calls a variety locally finite if every finitely generated algebra has finite cardinality, or equivalently, if every finitely generated free algebra has finite cardinality.
en.wikipedia.org
His most famous result is that the ring of invariants of binary forms of fixed degree is finitely generated.
en.wikipedia.org
A ring is called universally catenary if any finitely generated algebra over it is catenary.
en.wikipedia.org
This follows from the classification of finitely generated abelian groups.
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There is a natural similar question for iterating splittings of finitely generated groups over finite subgroups.
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Indeed, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem.
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This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced "k" -algebras.
en.wikipedia.org
Geometrically, finitely generated projective modules correspond to vector bundles over affine space, and free modules to trivial vector bundles.
en.wikipedia.org
If this is required only for finitely generated submodules, it is called semihereditary.
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