  We have studied the magnetic behavior of a 4d transition metal Ru monolayer
(ML) on different substrates and orientations. In the ground state, a Ru-ML is
expected to be ferromagnetic on Ag(001) and Au(001) with a magnetic moment
$(\mu) $ of 1.73 Bohr magnetons $(\mu_{B}) $ in both cases. On Cu(001), a Ru-ML
is not magnetic. In this paper, we study the magnetic behavior of a Ru-ML at
other orientations, i.e., (110) and (111). We found magnetism on Au (111), and
Ag (111) ($\mu\thicksim 1.3\mu_{B}$ for both) but no magnetic activity on a Cu
substrate in any orientation. Further, we found that on Ag(110), a Ru-ML is
ferromagnetic with $\mu =1.3\mu_{B}$. On Au(110), a Ru-ML is not magnetic.
Since on the (001) and (111) orientations a Ru-ML has about the same magnetic
activity on Ag as on Au, we found surprising the behavior in the (110)
orientation. We analyze it in detail in the final part of the paper. We found
that there exists a metastable ferromagnetic state, in this case, and that a
Ru-ML becomes ferromagnetic under a small expansion of the Au-lattice. This is
not the case for Cu.
