TITLE: What is an algebraic group over a noncommutative ring?
QUESTION [15 upvotes]: Let $R$ be a (noncommutative) ring.  (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.)  Then I think I know what "linear algebra in characteristic $R$" should be: it should be the study of the category $R\text{-bimod}$ of $(R,R)$-bimodules.  For example, an $R$-algebra on the one hand is a ring $S$ with a ring map $R \to S$.  But this is the same as a ring object in the $R\text{-bimod}$.  When $R$ is a field, we recover the usual linear algebra over $R$; in particular, when $R = \mathbb Z/p$, we recover linear algebra in characteristic $p$.
Suppose that $G$ is an algebraic group (or perhaps I mean "group scheme", and maybe I should say "over $\mathbb Z$"); then my understanding is that for any commutative ring $R$ we have a notion of $G(R)$, which is the group $G$ with coefficients in $R$.  (Probably there are some subtleties and modifications to what I just said.)

My question: What is the right notion of an algebraic group "in characteristic $R$"?

It's certainly a bit funny.  For example, it's reasonable to want $GL(1,R)$ to consist of all invertible elements in $R$.  On the other hand, in $R\text{-bimod}$, the group $\text{Aut}(R,R)$ consists of invertible elements in the center $Z(R)$.
Incidentally, I'm much more interested in how the definitions must be modified to accommodate noncommutativity than in how they must be modified to accommodate non-invertibility.  So I'm happy to set $R = \mathbb H$, the skew field of quaternions.  Or $R = \mathbb K[[x,y]]$, where $\mathbb K$ is a field and $x,y$ are noncommuting formal variables.

REPLY [7 votes]: Sorry for arriving here so late... I hope somebody will still notice my answer:
A possible definition would be to take a co-group in an appropriate category of noncommutative rings (or algebras), i.e. an object in such a category of noncommutative rings or algebras that represents a functor from this category into a category of groups (see also S. Carnahan's answer).
I know of several papers that follow this approach and study such co-groups, here is what I have been able to find:

Israel Berstein, On co-groups in the category of graded algebras,  Trans. Amer. Math. Soc. 115 (1965), 257–269.
Dan Voiculescu, Dual algebraic structures on operator algebras related to free products. J. Operator Theory 17 (1987), no. 1, 85–98. 
James J. Zhang, H-algebras. Adv. Math. 89 (1991), no. 2, 144–191. 
George Bergman, Adam Hausknecht,
Co-groups and co-rings in categories of associative rings.
Mathematical Surveys and Monographs, 45. American Mathematical Society, Providence, RI, 1996.
Benoit Fresse, Cogroups in algebras over an operad are free algebras, Comment. Math. Helv. 73 (1998) 637-676
Hiroshi Kihara, Cogroups in the category of connected graded algebras whose inverse and antipode coincide, arXiv:1303.7350
Loïc Foissy, Claudia Malvenuto, and Frederic Patras, B-infinity algebras, their enveloping algebras, and finite spaces. arXiv:1403.7488

Six papers and one book over a span of almost fifty years, citing each other rather sparsely (my list is certainly not complete, though). Compared to the vast literature on algebraic groups and group schemes, this does not seem to be a lot.
Question: Is there an explanation why noncommutative algebraic group theory (in the sense of the OP, maybe one should say noncocommutative?) is getting so little attention? E.g., lack of applications, technical difficulties, lack of interesting results?