Subdirectly Irreducible Rings Where Some Finite Conditions are Satisfied

It is proved that if a subdirectly irreducible ring R with the heart H=H² satisfies one of the following of conditions,then R≌M_n(F)(n≥2),where F=GF(P^m).(1) only finite non-zero nilpotent elements in H;(2) Only finite non-zero elements x in H,such as x^k=0;(where k>1);(3) only finite non-zero elements x in H, such as x²=0;(4) only finite non-zero right annihilators in H;(5) only finite non-zero left annilihilators in H;(6) only finite non-zero idmpotents in H,where e≠1.