  We establish a link between abelian regular subgroup of the affine group, and
commutative, associative algebra structures on the underlying vector space that
are (Jacobson) radical rings.
  As an application, we show that if the underlying field has positive
characteristic, then an abelian regular subgroup has finite exponent if the
vector space is finite-dimensional, while it can be torsion free if the
dimension is infinite.
  We also give an example of an abelian, regular subgroup of the affine group
over an infinite vector space, which intersects trivially the group of
translations.
