Geodesic
On $PI$-rings having a faithful module with Krull dimension
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 557-559 
V. T. Markov. On $PI$-rings having a faithful module with Krull dimension. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 557-559. http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a19/
@article{FPM_1995_1_2_a19,
     author = {V. T. Markov},
     title = {On $PI$-rings having a faithful module with {Krull} dimension},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {557--559},
     year = {1995},
     volume = {1},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a19/}
}
TY  - JOUR
AU  - V. T. Markov
TI  - On $PI$-rings having a faithful module with Krull dimension
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 1995
SP  - 557
EP  - 559
VL  - 1
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a19/
LA  - ru
ID  - FPM_1995_1_2_a19
ER  - 
%0 Journal Article
%A V. T. Markov
%T On $PI$-rings having a faithful module with Krull dimension
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 1995
%P 557-559
%V 1
%N 2
%U http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a19/
%G ru
%F FPM_1995_1_2_a19

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

The main result: if a $PI$-ring $R$ has a faithful left module $M$ with Krull dimension, then its prime radical $N$ is nilpotent. Moreover if the left modules $M$ and $N$ are finitely generated then $R$ has left Krull dimension which is equal to Krull dimension of the module $M$.

[1] Markov V. T., “Tochnye neterovy moduli nad $PI$-koltsami”, Abelevy gruppy i moduli, Vyp. 8, Tomsk, 1989, 99–103

[2] Amitsur S. A., Small L. W., “Finite-dimensional representations of $PI$-algebras”, J. Algebra, 133:2 (1990), 244–248 | DOI | MR | Zbl

[3] Gordon R., Robson J. C., Krull dimension, Mem. Amer. Math. Soc., 133, 1973 | MR | Zbl

[4] Procesi C., Rings with polynomial identities, Marcel Dekker, N. Y., 1973 | MR | Zbl

[5] Lenagan T. H., “Reduced rank in rings with Krull dimension”, Ring Theory, Antwerp 1978, Lecture Notes in Pure and Appl. Math., 51, Marcel Dekker, N. Y., 1979, 123–131 | MR

[6] Beidar K. I., Markov V. T., “Polupervichnoe $PI$-koltso, imeyuschee tochnyi modul s razmernostyu Krullya, yavlyaetsya koltsom Goldi”, Uspekhi mat. nauk, 48:6 (1993), 141–142 | MR | Zbl