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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   A M B   over left- and right-Artinian rings  A  and  B , respectively, is associated with the equivalent bimodule of translations   C M Z  , where  C  is the multiplicative ring of the bimodule   A M B   and  Z  is its center, and the relation between the quasi-Frobenius conditions for the bimodules   A M B   and   C M Z   is studied. It is demonstrated that, in the general case, the fact that   A M B   is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule   C M Z  . However, under some additional assumptions it can be shown that if   C M Z   is a quasi-Frobenius bimodule, then the bimodule   A M B   is quasi-Frobenius as well.

Artinian bimodule with quasi-Frobenius bimodule of translations