We investigate new compactifications of OM theory giving rise to a (3+1)-di
mensional open string theory with non-commutative x(0)-x(1) and x(2)-x(3) c
oordinates. The theory can be directly obtained by starting with a D3-brane
with parallel (near critical) electric and magnetic field components, in t
he presence of a RR scalar field. The magnetic parameter permits to interpo
late continuously between the x(0)-x(1) non-commutative open string theory
and the x(2)-x(3) spatial non-commutative U(N) super Yang-Mills theory. We
discuss SL(2, Z) transformations of this theory. Using the supergravity des
cription of the large-N limit, we also compute corrections to the quark-ant
iquark Coulomb potential arising in the NCOS theory.