Geodesic
Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring
Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 2, pp. 187-206 
A. Mekei. Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 2, pp. 187-206. http://geodesic.mathdoc.fr/item/FPM_2014_19_2_a9/
@article{FPM_2014_19_2_a9,
     author = {A. Mekei},
     title = {Varieties of associative rings containing a~finite ring that is nonrepresentable by a~matrix ring over a~commutative ring},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {187--206},
     year = {2014},
     volume = {19},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2014_19_2_a9/}
}
TY  - JOUR
AU  - A. Mekei
TI  - Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2014
SP  - 187
EP  - 206
VL  - 19
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/FPM_2014_19_2_a9/
LA  - ru
ID  - FPM_2014_19_2_a9
ER  - 
%0 Journal Article
%A A. Mekei
%T Varieties of associative rings containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2014
%P 187-206
%V 19
%N 2
%U http://geodesic.mathdoc.fr/item/FPM_2014_19_2_a9/
%G ru
%F FPM_2014_19_2_a9

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

In this paper, we give examples of infinite series of finite rings $B_v^{(m)}$, where $m\geq2$, $0\leq v\leq p-1$, and $p$ is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety $\operatorname{var}B_v^{(m)}$, where $m=2$, or $m-1=(p-1)k$, $k\geq1$, and $p\geq3$, or $p=2$, $m\geq3$, $0\leq v$, and $p$ is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order $p$.

[1] Algebraicheskie sistemy, Priprint No 647, SO AN SSSR, Vych. tsentr, Novosibirsk, 1986, 14–15

[2] Ananin Z. A., “Lokalno finitno approksimiruemye i lokalno predstavimye mnogoobraziya algebr”, Algebra i logika, 16:1 (1977), 3–23 | MR | Zbl

[3] Kublanovskii S. I., Lokalno finitno approksimiruemye i lokalno predstavimye mnogoobraziya assotsiativnykh kolets i algebr, Dep. v VINITI 1.12.82 No 6143-82

[4] Latyshev V. N., “Konechnaya baziruemost tozhdestv nekotorykh kolets”, Uspekhi mat. nauk, 32:4 (1977), 259–260 | MR | Zbl

[5] Lvov I. V., “O mnogoobraziyakh assotsiativnykh kolets. I”, Algebra i logika, 12:3 (1973), 269–297 | MR

[6] Lvov I. V., “O predstavlenii nilpotentnykh algebr matritsami”, Sib. mat. zhurn., 21:5 (1980), 158–161 | MR | Zbl

[7] Maltsev Yu. N., “O mnogoobraziyakh assotsiativnykh algebr”, Algebra i logika, 15:5 (1976), 579–589 | MR

[8] Maltsev Yu. N., “O predstavlenii konechnykh kolets matritsami nad kommutativnym koltsom”, Mat. sb., 128(170):3(11) (1985), 383–402 | MR | Zbl

[9] Mekei A., O mnogoobraziyakh, porozhdënnykh konechnymi assotsiativnymi koltsami, i svoistvakh ekstremalnosti i kritichnosti, Dis. $\dots$ dokt. fiz.-mat. nauk, M., 1995

[10] Mekei A., “O kritichnosti kolets endomorfizmov nekotorykh konechnykh abelevykh grupp”, Fundament. prikl. mat., 2:2 (1996), 449–482 | MR | Zbl

[11] Bergman G. M., “Some examples in PI ring theory”, Israel J. Math., 18:3 (1974), 257–277 | DOI | MR | Zbl

[12] Bergman G. M., Britten D. J., Lemire F. W., “Embedding rings in completed graded rings. III. Algebras over general $k$”, J. Algebra, 84:1 (1983), 42–61 | DOI | MR | Zbl

[13] Rowen L. H., Polynomial Identities in Ring Theory, Academic Press, New York, 1980 | MR | Zbl

[14] Wilson R. S., “On structure of finite rings. II”, Pacific J. Math., 51:1 (1974), 317–325 | DOI | MR