The magnetism and electronic structure of Ru monolayer with square lattice is investigated using the FLAPW band method. The dependence of total energies on the lattice constant was calculated for three magnetic states, i.e.,para-,ferro-, and antiferromagnetic ones. It was found that there is no energy difference between para-and antiferromagnetic states for all the lattice constant. The possibility of antiferromagnetism in square Ru monolayer is thus excluded. The ferromagnetic state is most stable for the lattice constants greater than 7.30 a.u. The energy minimum is found at the lattice constant of 6.53 a.u. Where it is paramagenetic. It is calculated that the magenetic moment is 2.49 ${MU}_B$ at 7.72 a.u., which is close to the lattice constant of Ag. The magnetic moment is almost saturated to be ${MU}_B$ at the lattice constant of 7.86 a.u.