# The number of standard forms of matrices over imaginary Euclidean quadratic rings with respect to the $(z,k)$–equivalence

### Abstract

The $(z,k)$--equivalence of matrices over imaginary Euclidean

quadratic rings is investigated. The classes of matrices over

these rings are selected for which the standard form with respect

to $(z,k)$--equivalence is uniquely defined and equal to the Smith

normal form. It is established that the number of standard forms

over imaginary Euclidean quadratic rings is finite. Bounds for a

number of standard forms are established.

### References

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*Matematychni Studii*,

*57*(2), 115-121. https://doi.org/10.30970/ms.57.2.115-121

Copyright (c) 2022 N. Ladzoryshyn, V. Petrychkovych

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Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.