  Classical primal-dual affine programming takes place over finite dimensional
real vector spaces. This results in beautiful duality theory, connecting the
optimal solu- tions of the primal maximization problem and the dual
minimization problems. These results include the Existence Duality Theorem,
which guarantees optimal solutions to any feasible bounded program; and the
Strong Duality Theorem, which implies that optimal solutions for primal and
dual programs must have the same objective value. In a common extension of
classical affine programming, we see that the Strong Duality does not hold when
ring of scalars is the integers. Extension of classical affine programming
results to ordered division rings are explored in. In this paper, we describe
the generalized setting of affine programming using ordered ring (not
necessarily division), and classify the rings for which the Existence Duality
Theorem or the Strong Duality Theorem fail.
