  A notion of one-dimensional formal ring is presented. It consists of a triple
$(A,\Phi,\Psi)$ where $A$ is a unital ring and $\Phi$ and $\Psi$ are two formal
power series in $2$ variables ${\Phi(x,y),\Psi(x,y)\in A\llbracket
x,y\rrbracket}$, the first one defining a one-dimensional formal group law over
$A$ and the second one providing a second composition law satisfying axiomatic
properties of compatibility with the first one. For a characteristic-zero ring
$A$, a large class of one-dimensional formal rings can be obtained by
constructing a new composition law, defined in terms of the group logarithm
associated with a given formal group law $\Phi(x,y)$, and associative and
distributive with respect to it. A natural $n$-dimensional generalization of
the previous construction is also proposed; curves on $n$-dimensional formal
rings are introduced. The higher-dimensional theory allows us to define a
generalization of the ring $W(A)$ of Witt vectors over a ring $A$, which is
recovered by means of a specific choice of the associated group logarithm. The
composition laws of our generalized Witt ring are defined in terms of an
underlying formal ring structure. Examples of formal rings related to
Hirzebruch's theory of genera are explicitly computed. Finally, we also propose
the examples of Euler's and Abel's formal rings.
