The definition of Krull dimension of a module over a ring $R$ in the sense of deviation of the poset of submodules ordered by inclusion may not coincide with the definition for non-Noetherian rings $R$ (though this is the case when $R$ is Noetherian). I want an example to show this difference.
Also, if all the prime ideals of a commutative (not necessarily Noetherian) ring $R$ are maximal could one deduce that the module $R_R$ has Krull dimension (in the sense of deviation of poset of its ideals)?
Thanks, in advance, for any cooperation!