On maximal nilpotent subrings of right Noetherian rings

Applying Hopkins's Theorem asserting that each unitary right Artinian ring is right Noetherian, G. Köthe and K. Shoda proved the following theorem (cf. Köthe [7], p. 360, Theorem 1 and p. 363, Theorem 5): If R is a unitary right Artinian ring, then the following statements hold:(i) Each nilpotent subring of R is contained in a maximal nilpotent subring of R.(ii) The intersection of all maximal nilpotent subrings of R is the maximal nilpotent twosided ideal of R.(iii) All maximal nilpotent subrings of R are conjugate.