Systems of linear equations that have the same solutions are row-equivalent if the equations are over $\mathbb{Z}_n$?

Question

I have seen that this is true if the linear equations are over <$\mathbb{R}$ and extensions>(Is there a converse for the following Theorem? Equivalent systems of linear equations have exactly the same solutions).

But is it true if the equations are over a ring $\mathbb{Z}_n$?