In this paper, as part of a project initiated by A. Mallios consisting
of exploring new horizons for Abstract Differential Geometry (`a la Mallios), [5,
6, 7, 8], such as those related to the classical symplectic geometry, we show that
essential results pertaining to biorthogonality in pairings of vector spaces do hold
for biorthogonality in pairings of A-modules. We single out that orthogonality
is reflexive for orthogonally convenient pairings of free A-modules of finite rank,
governed by non-degenerate A-morphisms, and where A is a PID (Corollary 3.8).
For the rank formula (Corollary 3.3), the algebra sheaf A is assumed to be a PID.
The rank formula relates the rank of an A-morphism and the rank of the kernel
(sheaf) of the same A-morphism with the rank of the source free A-module of the