Let $R$ be a finite ring and $F$ be an algebraically closed field in which $|R|$ is invertible. Does there exists an $F$-valued character $\chi$ of $(R, +)$ such that every character $\psi$ is of the form $\psi(a) = \chi(ab)$ for some $b \in R$? If not, does the statement hold when $R$ is commutative (or under any other reasonably nice assumption)?


This is true when $R$ is a finite field. I believe that the statement in this case follows from the non-degeneracy of the trace form over the prime field contained in $R$.


Any references would be really helpful.


Motivation: This appears when we study representation theory of unipotent groups. Suppose $U$ be the subgroup of unipotent lower triangular matrices in $GL\_n(R)$. Then elements of the form $E\_{\chi} = \sum\_{M \in U} \chi(M\_{2,1})M \in F[U]$ appear naturally where $\chi$ is an $F$-valued character of $(R, +)$. There is nothing special about $M\_{2,1}$, one can work with any closed "root subgroup". Now conjugating $E\_{\chi}$ by an element of the cartan subgroup yields $E\_{\chi(.b)}$ for some $b$ and so it becomes useful to know whether all characters can be obtained in this way. 



Action of certain endomorphisms on Pontriyagin dual
Call a character $\chi$ *(left) generating* if every character is of the form $\psi(a)=\chi(ab)$ for some $b\in R$. It turns out that, when $R$ is finite, a character is left generating if and only if it is right generating (every character has the form $\psi(a) = \chi(ba) $ for some $b\in R$). 


The answer to your question is as follows: 



> 
> A finite ring $R$ has a generating character if and only if $R$ is a finite [Frobenius ring](http://en.wikipedia.org/wiki/Quasi-Frobenius_ring). 
> 
> 
> 


For a proof, see, for example: 


*Thomas Honold*, MR 1831096 [**Characterization of finite Frobenius rings**](http://dx.doi.org/10.1007/PL00000451), *Arch. Math. (Basel)* **76** (2001), no. 6, 406--415. 


When the ring is finite and commutative, this is equivalent to $R$ having a multiplicity-free socle, or $R$ being a direct product of finite local rings with simple socle. (see *T. Y. Lam*, MR 1653294 [**Lectures on modules and rings**](http://dx.doi.org/10.1007/978-1-4612-0525-8) , Thm. 15.27, and more generally §§15, 16 for more characterizations and examples and non-examples.) 


Your question is quite natural and appears in other contexts. For example, it appears naturally when studying Gauss sums over arbitrary finite rings, as in a series of papers by Lamprecht (cited in Honold's paper), and in:  

*Fernando Szechtman*, MR 1916077 [**Quadratic Gauss sums over finite commutative rings**](http://dx.doi.org/10.1006/jnth.2001.2752), *J. Number Theory* **95** (2002), no. 1, 1--13. 


This property of finite Frobenius rings is also relevant to coding theory, see


*Jay A. Wood*, MR 1738408 [**Duality for modules over finite rings and applications to coding theory**](http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.3wood.pdf), *Amer. J. Math.* **121** (1999), no. 3, 555--575.


