Classification of finite rings: theory and algorithm

Mahmood Behboodi; Reza Beyranvand; Amir Hashemi; Hossein Khabazian

Czechoslovak Mathematical Journal (2014)

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

Abstract

top
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

top

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

top
  1. Chikunji, C. J., 10.1080/00927879908826747, Commun. Algebra 27 (1999), 5049-5081. (1999) Zbl0942.16027MR1709253DOI10.1080/00927879908826747
  2. Chikunji, C. J., On a class of rings of order p 5 , Math. J. Okayama Univ. 45 (2003), 59-71. (2003) Zbl1055.16023MR2038839
  3. Corbas, B., Williams, G. D., 10.1006/jabr.2000.8349, J. Algebra 231 (2000), 677-690. (2000) Zbl1017.16014MR1778165DOI10.1006/jabr.2000.8349
  4. Corbas, B., Williams, G. D., 10.1006/jabr.2000.8350, J. Algebra 231 (2000), 691-704. (2000) Zbl1017.16015MR1778166DOI10.1006/jabr.2000.8350
  5. Derr, J. B., Orr, G. F., Peck, P. S., 10.1016/0022-4049(94)00015-8, J. Pure Appl. Algebra 97 (1994), 109-116. (1994) MR1312757DOI10.1016/0022-4049(94)00015-8
  6. Eldridge, K. E., 10.2307/2314716, Am. Math. Mon. 75 (1968), 512-514. (1968) Zbl0157.07901MR0230772DOI10.2307/2314716
  7. Fine, B., 10.2307/2690742, Math. Mag. 66 (1993), 248-252. (1993) MR1240670DOI10.2307/2690742
  8. Gilmer, R., Mott, J., Associative rings of order p 3 , Proc. Japan Acad. 49 (1973), 795-799. (1973) Zbl0309.16015MR0369422
  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. (1980) MR0652254
  10. Raghavendran, R., A class of finite rings, Compos. Math. 22 (1970), 49-57. (1970) Zbl0212.37901MR0263876
  11. Raghavendran, R., Finite associative rings, Compos. Math. 21 (1969), 195-229. (1969) Zbl0179.33602MR0246905
  12. Shoda, K., 10.1007/BF01782346, Math. Ann. 102 (1929), 273-282 German. (1929) MR1512577DOI10.1007/BF01782346
  13. Wiesenbauer, J., 10.1007/BF01294779, Monatsh. Math. 78 (1974), 164-173 German. (1974) Zbl0286.16013MR0347893DOI10.1007/BF01294779
  14. Wilson, R. S., On the structure of finite rings, Compos. Math. 26 (1973), 79-93. (1973) Zbl0248.16009MR0320065
  15. Wilson, R. S., 10.2140/pjm.1974.51.317, Pac. J. Math. 51 (1974), 317-325. (1974) Zbl0317.16009MR0352169DOI10.2140/pjm.1974.51.317
  16. Wilson, R. S., 10.2140/pjm.1974.53.643, Pac. J. Math. 53 (1974), 643-649. (1974) Zbl0317.16008MR0369423DOI10.2140/pjm.1974.53.643

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.