We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut (G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.

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Anderson and Ohm have introduced valuations of monoid rings k[Γ] where k is a field and Γ a cancellative torsion-free commutative monoid. We study the residue class fields in question and solve a problem concerning the pure transcendence of the residue fields.