On equivalence classes of matrices over a finite field of odd characteristic
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1200-1210.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we classified up to isomorphism all finite local rings $R$ with Jacobson radical $J$ and conditions: $$\mathrm{char} R\neq 2,\ R/J=F\subseteq Z(R),\ {\dim_F J/J^2=2},\ {\dim_F J^2=3},\ {J^3=0}.$$
Keywords: finite rings, local rings.
@article{SEMR_2023_20_2_a12,
     author = {E. V. Zhuravlev},
     title = {On equivalence classes of matrices over a finite field of odd characteristic},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1200--1210},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a12/}
}
TY  - JOUR
AU  - E. V. Zhuravlev
TI  - On equivalence classes of matrices over a finite field of odd characteristic
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2023
SP  - 1200
EP  - 1210
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a12/
LA  - ru
ID  - SEMR_2023_20_2_a12
ER  - 
%0 Journal Article
%A E. V. Zhuravlev
%T On equivalence classes of matrices over a finite field of odd characteristic
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2023
%P 1200-1210
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a12/
%G ru
%F SEMR_2023_20_2_a12
E. V. Zhuravlev. On equivalence classes of matrices over a finite field of odd characteristic. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1200-1210. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a12/

[1] A.D. Porter, “Orthogonal similarity in a finite field”, Math. Nachr., 40 (1969), 327–331 | DOI | MR | Zbl

[2] A.D. Porter, J. Adams, “Similarity and orthogonal similarity in a finite field”, Duke Math. J., 35 (1968), 519–524 | DOI | MR | Zbl

[3] M. Newman, Integral Matrices, Academic Press, New York-London, 1972 | MR | Zbl

[4] W.C. Waterhouse, “The number of songruence slasses in $M_n(F_q)$”, Finite Fields and Their Applications, 1:1 (1995), 57–63 | DOI | MR | Zbl

[5] P.S. Bremser, “Congruence classes of matrices in $GL_2(F_q)$”, Discrete Math., 118:1-3 (1993), 243–249 | DOI | MR | Zbl

[6] G.D. Williams, “Congruence of $(2\times 2)$ matrices”, Discrete Math., 224:1-3 (2000), 293–297 | DOI | MR | Zbl

[7] B. Gorbas, G.D. Williams, “Matrix representatives for three-dimensional bilinear forms over finite fields”, Discrete Math., 185:1-3 (1998), 51–61 | DOI | MR | Zbl

[8] B. Gorbas, G.D. Williams, “Congruence classes in $M_3(F_q)$ ($q$ odd)”, Discrete Math., 219:1-3 (2000), 37–47 | DOI | MR | Zbl

[9] B. Gorbas, G.D. Williams, “Congruence classes in $M_3(F_q)$ ($q$ even)”, Discrete Math., 257:1 (2002), 15–27 | DOI | MR | Zbl

[10] B. Gorbas, G.D. Williams, “Rings of order $p^5$. I: Nonlocal rings”, J. Algebra, 231:2 (2000), 677–690 | DOI | MR | Zbl

[11] B. Gorbas, G.D. Williams, “Rings of order $p^5$. II: Local rings”, J. Algebra, 231:2 (2000), 691–704 | DOI | MR | Zbl

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

[13] B. Gorbas, G.D. Williams, “Gongruence of two-dimensional subspaces in $M_2(K)$ (characteristic $\neq 2$)”, Pac. J. Math., 188:2 (1999), 225–235 | DOI | MR | Zbl

[14] B. Gorbas, G.D. Williams, “Gongruence of two-dimensional subspaces in $M_2(K)$ (characteristic 2)”, Pac. J. Math., 188:2 (1999), 237–249 | DOI | MR | Zbl

[15] C.J. Chikunji, “Using Matlab to solve a classification problem in finite rings”, $2^{nd}$ international conference on the teaching of mathematic (Greece), 2002 http://www.math.uoc.gr/ĩctm2/Proceedings/pap252.pdf

[16] V.P. Elizarov, Finite rings, Gelios ARV, M., 2006

[17] V.G. Antipkin, V.P. Elizarov, “Rings of order $p^3$”, Sib. Math. J., 23 (1982), 457–464 | DOI | MR | Zbl

[18] V.P. Elizarov, Nonnilpotent finite rings, The manuscript of the dep. in the SB of the USSR Academy of Sciences, No 1472, editorial board of the Siberian Mathematical Journal, 1985 \

[19] J.B. Derr, G.F. Orr, P.S. Peck, “Noncommutative rings of order $p^4$”, J. Pure Appl. Algebra, 97:2 (1994), 109–116 | DOI | MR | Zbl

[20] E.V. Zhuravlev, “Local rings of order $p^6$ with 4-nilpotent Jacobson radical”, Sib. Èlectron. Mat. Izv., 3 (2006), 15–59 | MR | Zbl

[21] E.V. Zhuravlev, “On the classification of finite commutative local rings”, Sib. Èlectron. Mat. Izv., 12 (2015), 625–638 | MR | Zbl

[22] C.J. Chikunji, “On a class of finite rings”, Commun. Algebra, 27:10 (1999), 5049–5081 | DOI | MR | Zbl

[23] R. Raghavendran, “Finite associative rings”, Compos. Math., 21 (1969), 195–229 | MR | Zbl