# Classification of finite rings: theory and algorithm

• Volume: 64, Issue: 3, page 641-658
• ISSN: 0011-4642

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## Abstract

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An interesting topic in the ring theory is the classification of finite rings. Although rings of certain orders have already been classified, a full description of all rings of a given order remains unknown. The purpose of this paper is to classify all finite rings (up to isomorphism) of a given order. In doing so, we introduce a new concept of quasi basis for certain type of modules, which is a useful computational tool for dealing with finite rings. Then, using this concept, we give structure and isomorphism theorems for finite rings and state our main result to classify (up to isomorphism) the finite rings of a given order. Finally, based on these results, we describe an algorithm to calculate the structure of all such rings. We have implemented our new algorithm in Maple, and we apply it to an example.

## How to cite

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Behboodi, Mahmood, et al. "Classification of finite rings: theory and algorithm." Czechoslovak Mathematical Journal 64.3 (2014): 641-658. <http://eudml.org/doc/262155>.

@article{Behboodi2014,
abstract = {An interesting topic in the ring theory is the classification of finite rings. Although rings of certain orders have already been classified, a full description of all rings of a given order remains unknown. The purpose of this paper is to classify all finite rings (up to isomorphism) of a given order. In doing so, we introduce a new concept of quasi basis for certain type of modules, which is a useful computational tool for dealing with finite rings. Then, using this concept, we give structure and isomorphism theorems for finite rings and state our main result to classify (up to isomorphism) the finite rings of a given order. Finally, based on these results, we describe an algorithm to calculate the structure of all such rings. We have implemented our new algorithm in Maple, and we apply it to an example.},
author = {Behboodi, Mahmood, Beyranvand, Reza, Hashemi, Amir, Khabazian, Hossein},
journal = {Czechoslovak Mathematical Journal},
keywords = {classification of finite ring; finite abelian group; quasi base; classification of finite ring; finite abelian group; quasi base},
language = {eng},
number = {3},
pages = {641-658},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Classification of finite rings: theory and algorithm},
url = {http://eudml.org/doc/262155},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Behboodi, Mahmood
AU - Beyranvand, Reza
AU - Hashemi, Amir
AU - Khabazian, Hossein
TI - Classification of finite rings: theory and algorithm
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 641
EP - 658
AB - An interesting topic in the ring theory is the classification of finite rings. Although rings of certain orders have already been classified, a full description of all rings of a given order remains unknown. The purpose of this paper is to classify all finite rings (up to isomorphism) of a given order. In doing so, we introduce a new concept of quasi basis for certain type of modules, which is a useful computational tool for dealing with finite rings. Then, using this concept, we give structure and isomorphism theorems for finite rings and state our main result to classify (up to isomorphism) the finite rings of a given order. Finally, based on these results, we describe an algorithm to calculate the structure of all such rings. We have implemented our new algorithm in Maple, and we apply it to an example.
LA - eng
KW - classification of finite ring; finite abelian group; quasi base; classification of finite ring; finite abelian group; quasi base
UR - http://eudml.org/doc/262155
ER -

## References

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