[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 R is a commutative ring then the matrix ring M_n(R) is an associative algebra.

Fails if R is non-commutative: for then we need not have r(AB) = A(rB).