On finite strongly critical rings (Q2212710)

The authors consider properties of finite (associative) rings. Recall that a finite ring \(A\) is critical if \(A\) does not lie in the variety generated by all of its proper homomorphic images. It is well known that a critical finite ring is a subdirectly irreducible, and its order is \(p^n\) for some prime \(p\). A ring \(R\) of order \(n\) is strongly critical if it does not belong to the variety generated by all rings of order less than \(n\). The authors prove the following results. If \(R\) is a simple finite ring then it is strongly critical. The same conclusion holds if \(R\) is critical and of order \(p^2\) where \(p\) is prime. They give an example of a ring of order 8 which is critical but not strongly critical. Moreover, if \(R\) is a finite ring and \(M_n(R)\) is strongly critical then \(R\) itself is strongly critical. The converse also holds. Finally if \(R\) is finite, and \(R/J(R)\cong M_n(GF(q))\) , and if \(J(R)\) is strongly critical then \(R\) itself is strongly critical. Here \(J(R)\) is the radical of \(R\), and \(GF(q)\) is the field with \(q\) elements.