I am an undergrad student who wants to know about the representation theory over


arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular 


representation theory.)


In particular, I would like to study the representation theory of modular Heisenberg groups


(entries in a finite field)over finite rings. I will start with the homomorphisms from 


this Heisenberg groups to the ring of integers and the finite ring of integers.


Actually, I mean that does it make sense "over finite ring".


If it does, I would like to know good references and brief explanations.



Does it make sense that "Representations of groups over finite ring" ?
EDIT: Everything below the line is from before the OP changed his question. It was attempting to study linear maps $\rho: R \rightarrow GL(V)$ for $R$ a finite ring and $V$ a vector space. Now that the question has been cleared up, we see that the OP desired to study $\rho: G \rightarrow GL(R)$ for $G$ a group and $R$ a ring. Here's my attempt to see this question resolved, following a recommended procedure in [meta](http://mathoverflow.tqft.net/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/#Item_0) (link broken, thread archived [here](https://msleziak.com/mathoverflow/tea/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/#Item_0)) by linking to the references given in the comments above and making my answer CW so I don't get rep for this.


[S. Carnahan](https://mathoverflow.net/questions/69019/does-it-make-sense-that-representations-of-groups-over-finite-ring#comment174481_69019) mentioned Curtis and Reiner's text *[Representation theory of finite groups and associative algebras.](https://books.google.com/books?id=RKwjeZKMr8oC "zbMATH review at https://zbmath.org/?q=an:1093.20003")* A few moments of browsing shows me that page 30 has the first definition of representation and one can see from there how to develop the theory the OP sought.


[Dan Ramras](https://mathoverflow.net/questions/69019/does-it-make-sense-that-representations-of-groups-over-finite-ring#comment174415_69019) recommends Dave Benson's book *[Representations and Cohomology: Basic representation theory of finite groups and associative algebras.](https://books.google.com/books?id=cezF63340GAC "zbMATH review at https://zbmath.org/?q=an:0718.20001")* Note that this book comes in two parts, and the OP almost certainly wants the first part since the second is highly topological and focuses on group cohomology.




---


I did some googling and found a few references. First, [this](http://www.springerlink.com/content/t0584663311474l7/) article reduces a problem the authors are interested in to a well known problem in the representation theory of finite rings. At the end are some references to that problem and those should include the necessary background you need to get into this subject.


Another reference is the MO post [Lecture notes on representations of finite groups](https://mathoverflow.net/questions/6394/lecture-notes-on-representations-of-finite-groups), which includes as many freely available texts on representation theory of groups as could be found. Perhaps one of them discusses or mentions representation theory of rings as well.


In case none of those references pan out, here is perhaps a place to start your thinking. A representation of a group is $\rho: G \rightarrow GL(\mathbb{C})$ and is equivalent to a module over the group ring $\mathbb{C}[G]$. A representation of a ring would need to be $\rho: R\rightarrow GL(\mathbb{C})$ and should be equivalent to a module $M$ over a more general ring. We can form $\det \circ \rho$: from the units of $R$ to $\mathbb{C}$, and this restricts the form $M$ can take.


