Braman [B08] described a construction where third-order tensors are exactly
the set of linear transformations acting on the set of matrices with vectors as
scalars. This extends the familiar notion that matrices form the set of all
linear transformations over vectors with real-valued scalars. This result is
based upon a circulant-based tensor multiplication due to Kilmer et al.
[KMP08]. In this work, we generalize these observations further by viewing this
construction in its natural framework of group rings.The circulant-based
products arise as convolutions in these algebraic structures. Our
generalization allows for any abelian group to replace the cyclic group, any
commutative ring with identity to replace the field of real numbers, and an
arbitrary order tensor to replace third-order tensors, provided the underlying
ring is commutative.