On the nilpotency of the Jacobson radical for noetherian rings By

Abstract. Let R be a right noetherian ring of Krull dimension k and denote by J its Jacobson radical. Using a new definition of the powers Ja (a any ordinal) it is shown that Jwkn 0 for some natural number n. Let R be a right noetherian ring and denote by J its Jacobson radical. A well-known result of Jacobson [2] states that J\u85n 0 for some ordinal n, where for any ordinal a one defines J a J\u85bJ if a b 1, and J\u85a T g<a J g if a is a limit ordinal. In this note we follow Prest [5] and define for any ordinal a the power Ia of an ideal I as follows: If a is a finite ordinal, then Ia Pr i1 xi1 xia j xij 2 I; r 2 N. If a is a limit ordinal, let Ia T g<a Ig, and if a is an infinite non-limit ordinal, so uniquely of the form a b n for some limit ordinal b and some natural number n ^ 1, let Ia \u85 Ibn1. With this definition we obtain the following bound for the index of nilpotency of J: Theorem. Let R be a right noetherian ring of Krull dimension k. Then Jwkn 0 for some natural number n. In [4], Krause and Lenagan introduced another radical series \u85Jaa by definin