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I recently read about the GEM equations, which look very much like the Maxwell's equations.

Does this mean the behavior of mass is like the the behavior of the electric charge?

So for example if you spin a ring so you have ring "mass-current" do you get a "gravitomagnet"? If you get two of such spinning rings can you make them repel each other, like electromagnets do when the currect is opposite?

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The Gravitoelectromagnetic equations are exactly the same as Maxwell's equations with $\epsilon_0$ replaced by $(-4\,\pi\,G)^{-1}$, so, to the extent that the GEM equations approximate the Einstein field equations, the behavior of mass is very much like that of electric charge. Here are the differences:

  1. The minus sign in the "gravitoelectric constant" $(-4\,\pi\,G)^{-1}$ cannot be spirited away by a co-ordinate transformation and this has the physical meaning that, whereas like electrical charges repel, like masses gravitationally attract. So in your spinning wheel example, two wheels spinning in opposite directions about the same axis will have attractive gravitomagnetic force component: further to their static gravitational attraction.

  2. The analogue of the Lorentz force has a factor of four in it: $\vec{F} = m\,(\vec{E}+4\,\vec{v}\times\vec{B})$;

  3. The relativistic mass $m$ in GEM is not Lorentz invariant, whereas the electric charge is: the latter is a true scalar;

Look at the Wikipedia Gravitoelectromagnetism page, especially under the "Scaling of fields" section and also the "Lack of Invariance" Section.

Witness, though, that "charge" and "mass" from the standpoint of any physical theory are very alike at the outset, simply by being a "coupling" constant between something and a field. You can derive GEM along the lines that Laplace thought about: we begin with Newton's gravitation, whose inverse square law is analogous to electrostatics, and add a delay in the propagation of the effects of gravity but we do it in a Lorentz covariant (aside from the source, as pointed out in 3. above) way; Laplace didn't know about Lorentz covariance and so his naive theory makes planetary orbits grossly unstable as I talk about in my answer here and so the great man sadly went wrong. Just think, he might have foretold the apsidal precession of Mercury's orbit! The inverse square law can be thought of as resulting from the symmetrical spreading of any radial field in three dimensions, so it's not too surprising that the theories are so alike. Ultimately the qualitative difference between GEM and the Einstein Field Equations is that the sources are tensors of rank one and two respectively (four-current in Maxwell's equations/ GEM and Stress Energy Tensor in the EFE).

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