Abelian regular subgroups of the affine group and radical rings

We establish a correspondence 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-dimensional vector space, which intersects trivially the group of translations. This theory has found applications to the study of the group generated by the round functions of an AES-like cipher (in the paper "An application of the O'Nan-Scott theorem to the group generated by the round functions of an AES-like cipher" by the same authors); and to some questions related to abelian Hopf Galois structures (in the paper "Abelian Hopf Galois structures on prime-power Galois field extensions" by S. C. Featherstonhaugh, L. N. Childs and the first author).