On finite strongly critical rings

Yu. N. Maltsev ; E. V. Zhuravlev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1722-1729

Résumé

In the present paper, some properties of strongly critical rings are investigated. It is proved that every simple finite ring and each critical ring of order $ p ^ 2 $ ($ p $ is a prime) are strongly critical. There is an example of critical ring of order 8 which is not strongly critical. It is also proved that if $ R $ is a finite ring and $ M_n (R) $ is a strongly critical ring, then $ R $ is a strongly critical ring. For rings with unity, it is proved that: 1) if $ R $ is a finite ring, $ R / J (R) = M_n (GF (q)) $ and $ J (R) $ is a strongly critical ring, then $ R $ is a strongly critical ring; 2) $R$ is strongly critical ring iff $M_n(R)$ is a strongly critical ring (for any $n\geq 1$).

Détail

Keywords: finite ring, critical ring, strongly critical ring.