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Artinian bimodule with quasi-Frobenius bimodule of translations

  • Aleksandr A. Nechaev and Vadim N. Tsypyschev EMAIL logo

Abstract

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


Note: Originally published in Diskretnaya Matematika (2017) 29, №3, 70–91 (in Russian).


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Received: 2016-10-10
Revised: 2017-07-03
Published Online: 2019-04-12
Published in Print: 2019-04-24

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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