I am trying to understand the relationship between Lie algebras, Lie groups, and Lie modules. I know that a Lie algebra is a vector space with a bilinear, antisymmetric, and Jacobi-satisfying bracket operation, and that a Lie group is a smooth manifold with a group structure that is compatible with the smooth structure. I also know that a Lie algebra can be seen as the tangent space at the identity of a Lie group.
A Lie algebra is defined by a field, and a Lie module is defined by a ring. I wonder if there is a similar notion of a Lie module for a Lie group, and if so, what is the name and the definition of it? Is there a correspondence between Lie modules and these objects, analogous to the correspondence between Lie algebras and Lie groups?
(My problems is to define an object which in its tangent is a Lie module (as defined below), as a Lie group is in its tangent a Lie algebra.)
EDIT :
I define a Lie module as :
A Lie Module is a module $\mathfrak {g}$ over some ring $R$ together with a binary operation together with a binary operation $\displaystyle [\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$ called the Lie bracket satisfying bilinearity, alternativity, and the Jacobi identity.
This is not the common definition of a Lie module. Usually, a Lie module (aka Lie algebra representation) is defined as a module on which a Lie algebra acts (in a suitable way). A Lie algebra can be defined over any commutative ring (in fact, in every linear tensor category). This is all entirely analogous to associative algebras.
I think your question is rather what Lie groups over arbitrary commutative rings are. But Torsten already answered that in the comments. Lie groups are group objects in the category of smooth manifolds, so you essentially ask what manifolds over arbitrary commutative rings are. Well, one answer are (smooth) schemes. So the corresponding "Lie" objects are the (smooth) group schemes. The notion of tangent spaces is also defined for schemes.