| Title:
|
Classification of finite rings: theory and algorithm (English) |
| Author:
|
Behboodi, Mahmood |
| Author:
|
Beyranvand, Reza |
| Author:
|
Hashemi, Amir |
| Author:
|
Khabazian, Hossein |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
641-658 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
classification of finite ring |
| Keyword:
|
finite abelian group |
| Keyword:
|
quasi base |
| MSC:
|
16P10 |
| MSC:
|
16Z05 |
| MSC:
|
68W30 |
| idZBL:
|
Zbl 06391517 |
| idMR:
|
MR3298552 |
| DOI:
|
10.1007/s10587-014-0124-7 |
| . |
| Date available:
|
2014-12-19T15:59:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144050 |
| . |
| Reference:
|
[1] Chikunji, C. J.: On a class of finite rings.Commun. Algebra 27 (1999), 5049-5081. Zbl 0942.16027, MR 1709253, 10.1080/00927879908826747 |
| Reference:
|
[2] Chikunji, C. J.: On a class of rings of order $p^5$.Math. J. Okayama Univ. 45 (2003), 59-71. Zbl 1055.16023, MR 2038839 |
| Reference:
|
[3] Corbas, B., Williams, G. D.: Rings of order $p^5$ I: Nonlocal rings.J. Algebra 231 (2000), 677-690. Zbl 1017.16014, MR 1778165, 10.1006/jabr.2000.8349 |
| Reference:
|
[4] Corbas, B., Williams, G. D.: Rings of order $p^5$ II: Local rings.J. Algebra 231 (2000), 691-704. Zbl 1017.16015, MR 1778166, 10.1006/jabr.2000.8350 |
| Reference:
|
[5] Derr, J. B., Orr, G. F., Peck, P. S.: Noncommutative rings of order $p^4$.J. Pure Appl. Algebra 97 (1994), 109-116. MR 1312757, 10.1016/0022-4049(94)00015-8 |
| Reference:
|
[6] Eldridge, K. E.: Orders for finite noncommutative rings with unity.Am. Math. Mon. 75 (1968), 512-514. Zbl 0157.07901, MR 0230772, 10.2307/2314716 |
| Reference:
|
[7] Fine, B.: Classification of finite rings of order $p^2$.Math. Mag. 66 (1993), 248-252. MR 1240670, 10.2307/2690742 |
| Reference:
|
[8] Gilmer, R., Mott, J.: Associative rings of order $p^3$.Proc. Japan Acad. 49 (1973), 795-799. Zbl 0309.16015, MR 0369422 |
| Reference:
|
[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 0652254 |
| Reference:
|
[10] Raghavendran, R.: A class of finite rings.Compos. Math. 22 (1970), 49-57. Zbl 0212.37901, MR 0263876 |
| Reference:
|
[11] Raghavendran, R.: Finite associative rings.Compos. Math. 21 (1969), 195-229. Zbl 0179.33602, MR 0246905 |
| Reference:
|
[12] Shoda, K.: Über die Einheitengruppe eines endlichen Ringes.Math. Ann. 102 (1929), 273-282 German. MR 1512577, 10.1007/BF01782346 |
| Reference:
|
[13] Wiesenbauer, J.: Über die endlichen Ringe mit gegebener additiver Gruppe.Monatsh. Math. 78 (1974), 164-173 German. Zbl 0286.16013, MR 0347893, 10.1007/BF01294779 |
| Reference:
|
[14] Wilson, R. S.: On the structure of finite rings.Compos. Math. 26 (1973), 79-93. Zbl 0248.16009, MR 0320065 |
| Reference:
|
[15] Wilson, R. S.: On the structure of finite rings II.Pac. J. Math. 51 (1974), 317-325. Zbl 0317.16009, MR 0352169, 10.2140/pjm.1974.51.317 |
| Reference:
|
[16] Wilson, R. S.: Representations of finite rings.Pac. J. Math. 53 (1974), 643-649. Zbl 0317.16008, MR 0369423, 10.2140/pjm.1974.53.643 |
| . |
