TITLE: Chevalley Groups over an arbitrary ring.
QUESTION [6 upvotes]: My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the computations and I noticed that the computations works over any commutative $\mathbb{Z}$-algebra. Why these groups are not introduced in the general format? Of course if these groups are defined over rings then they are not necessarily simple. Is it the reason why most people want to define them over fields?    
Just to give a general definition of Chevalley groups over rings let me review the construction.
Let $L$ be a finite dimension complex simple Lie algebra with a Chevalley basis 
$$\{ e_r: r\in\Pi; e_r: r\in\Phi \},$$
where $\Pi$ is basis for the root system $\Phi$. Pick $\zeta\in\mathbb{C}$. Therefore 
$$
\exp(\zeta ad_{e_r}),
$$
is an Lie-algebra automorphism of $L$. One can observe that the entries of the matrix associated to $\exp(\zeta ad_{e_r})$, denoted by $A_r(\zeta)$, with respect to a Chevalley basis, are of the form $a\zeta^l$ where $a\in\mathbb{Z}$ and $l\in\mathbb{Z}_{\geq 0}$. Now let $B$ be a commutative $Z$ algebra with the structure map $\rho: \mathbb{Z}\to B$. Pick $b\in B$, then we consider the matrix $\overline{A_r(b)}$ which is obtained by transforming the entry of $a\zeta^l$ of the matrix $A_r(\zeta)$ into $\rho(a)b^l$. Then One can consider the linear transformation, denoted by $\overline{x_r(b)}$, obtained by $\overline{A_r(b)}$ on $L(B):=L(\mathbb{Z})\otimes_{\mathbb{Z}}B$. One can show that $\overline{x_r(b)}$ is indeed a Lie-algebra automorphism  of $L(B)$. Then the Chevalley group $G_{ad}(B,\Phi)$, is defined by the subgroup of $GL(L(B))$ generated by $\overline{x_r(b)}$.   
Is there a problem in this construction that I am not considering? 
Many thanks for your answers.

REPLY [6 votes]: A couple of further clarifications, to supplement the extensive answer by marguax and the many comments:

Chevalley's influential 1955 paper was mainly concerned with finding a uniform approach to most of the known simple finite groups of Lie type (supplemented soon afterward by the introduction of twisted groups as well as the groups of Suszuki and Ree).   The original "Chevalley group" was defined as a group of automorphisms of a certain Lie algebra, generated by unipotent elements; this group is simple over almost all fields (except a few very small ones).   The Lie algebra here is obtained by first reducing mod $p$  a Chevalley $\mathbb{Z}$-basis of a simple Lie algebra over $\mathbb{C}$, then tensoring with a field of prime characteristic $p$.

A subtle point here is that the "Chevalley Lie algebra" obtained over an algebraically closed field is actually the Lie algebra of the corresponding simply connected algebraic group, whereas the Chevalley groups themselves are closer to the adjoint groups.
Anyway, Steinberg in his 1967-68 Yale lectures broadened the notion of "Chevalley group" by using other faithful Lie algebra representations, to include special linear groups and the like.

In fact, there is a huge amount of literature about Chevalley groups over (commutative) rings.   Here the ideal structure of the ring contributes to normal subgroup structure in the group, so the groups are typically non-simple.   But they do come up naturally in algebraic K-theory, including the study of the congruence subgroup problem.    N.A. Vavilov and many others have written extensively about such groups.   There is less textbook literature along these lines, except for a few books on algebraic K-theory including one by Hahn and O'Meara.   And the entire subject becomes quite technical.

ADDED: Concerning the more sophisticated viewpoint of Chevalley-Demazure group schemes, there is an important recent paper by Lusztig (not yet freely available online) Study of a $\mathbf{Z}$-form of the coordinate ring of a reductive group.   The arXiv preprint is here.