[alg]

Subgroups of finitely-generated abelian groups are finitely-generated.

Not necessarily true for finitely-generated non-abelian groups.

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[alg]

Subgroups of finitely-generated abelian groups are finitely-generated.

Not necessarily true for finitely-generated non-abelian groups.

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[alg]

If is a *commutative* ring then the matrix ring is an associative algebra.

Fails if is non-commutative: for then we need not have .