Artinian bimodule with quasi-Frobenius bimodule of translations

The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule $_AM_B$ over left- and right-Artinian rings $A$ and $B$, respectively, is associated with the equivalent bimodule of translations $_CM_Z$, where $C$ is the multiplicative ring of the bimodule $_AM_B$ and $Z$ is its center, and the relation between the quasi-Frobenius conditions for the bimodules $_AM_B$ and $_CM_Z$ is studied. It is demonstrated that, in the general case, the fact that $_AM_B$ is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule $_CM_Z$. However, under some additional assumptions it can be shown that if $_CM_Z$ is a quasi-Frobenius bimodule, then the bimodule $_AM_B$ is quasi-Frobenius as well. } \keywords{ Quasi-Frobenius bimodule, Artinian ring, multiplicative ring, bimodule of translatios, linear recurrent sequence