Geodesic
Classification of finite rings: theory and algorithm
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 641-658 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-014-0124-7
Classification : 16P10, 16Z05, 68W30
Keywords: classification of finite ring; finite abelian group; quasi base 
@article{10_1007_s10587_014_0124_7,
     author = {Behboodi, Mahmood and Beyranvand, Reza and Hashemi, Amir and Khabazian, Hossein},
     title = {Classification of finite rings: theory and algorithm},
     journal = {Czechoslovak Mathematical Journal},
     pages = {641--658},
     year = {2014},
     volume = {64},
     number = {3},
     doi = {10.1007/s10587-014-0124-7},
     mrnumber = {3298552},
     zbl = {06391517},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0124-7/}
}
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
SP  - 641
EP  - 658
VL  - 64
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0124-7/
DO  - 10.1007/s10587-014-0124-7
LA  - en
ID  - 10_1007_s10587_014_0124_7
ER  - 
%0 Journal Article
%A Behboodi, Mahmood
%A Beyranvand, Reza
%A Hashemi, Amir
%A Khabazian, Hossein
%T Classification of finite rings: theory and algorithm
%J Czechoslovak Mathematical Journal
%D 2014
%P 641-658
%V 64
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0124-7/
%R 10.1007/s10587-014-0124-7
%G en
%F 10_1007_s10587_014_0124_7
Behboodi, Mahmood; Beyranvand, Reza; Hashemi, Amir; Khabazian, Hossein. Classification of finite rings: theory and algorithm. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 641-658. doi: 10.1007/s10587-014-0124-7

[1] Chikunji, C. J.: On a class of finite rings. Commun. Algebra 27 (1999), 5049-5081. | DOI | MR | Zbl

[2] Chikunji, C. J.: On a class of rings of order $p^5$. Math. J. Okayama Univ. 45 (2003), 59-71. | MR | Zbl

[3] Corbas, B., Williams, G. D.: Rings of order $p^5$ I: Nonlocal rings. J. Algebra 231 (2000), 677-690. | DOI | MR | Zbl

[4] Corbas, B., Williams, G. D.: Rings of order $p^5$ II: Local rings. J. Algebra 231 (2000), 691-704. | DOI | MR | Zbl

[5] Derr, J. B., Orr, G. F., Peck, P. S.: Noncommutative rings of order $p^4$. J. Pure Appl. Algebra 97 (1994), 109-116. | DOI | MR

[6] Eldridge, K. E.: Orders for finite noncommutative rings with unity. Am. Math. Mon. 75 (1968), 512-514. | DOI | MR | Zbl

[7] Fine, B.: Classification of finite rings of order $p^2$. Math. Mag. 66 (1993), 248-252. | DOI | MR

[8] Gilmer, R., Mott, J.: Associative rings of order $p^3$. Proc. Japan Acad. 49 (1973), 795-799. | MR | Zbl

[9] Lidl, R., Wiesenbauer, J.: Ring Theory and Applications. Foundations and Examples of Application in Coding Theory and in Genetics. Textbooks for Mathematics Akademische Verlagsgesellschaft, Wiesbaden (1980), German. | MR

[10] Raghavendran, R.: A class of finite rings. Compos. Math. 22 (1970), 49-57. | MR | Zbl

[11] Raghavendran, R.: Finite associative rings. Compos. Math. 21 (1969), 195-229. | MR | Zbl

[12] Shoda, K.: Über die Einheitengruppe eines endlichen Ringes. Math. Ann. 102 (1929), 273-282 German. | DOI | MR

[13] Wiesenbauer, J.: Über die endlichen Ringe mit gegebener additiver Gruppe. Monatsh. Math. 78 (1974), 164-173 German. | DOI | MR | Zbl

[14] Wilson, R. S.: On the structure of finite rings. Compos. Math. 26 (1973), 79-93. | MR | Zbl

[15] Wilson, R. S.: On the structure of finite rings II. Pac. J. Math. 51 (1974), 317-325. | DOI | MR | Zbl

[16] Wilson, R. S.: Representations of finite rings. Pac. J. Math. 53 (1974), 643-649. | DOI | MR | Zbl

Cité par Sources :