RINGS WITH A BOUNDED NUMBER OF GENERATORS FOR RIGHT IDEALS

IF R is a commutative Noetherian ring in which every right ideal can be generated by a bounded number of elements, then it is well known and easy to prove that R has Krull dimension at most one [C]. In a noncommutative setting the situation is, however, less clear. Certainly no such result is possible for right Noetherian rings, as Jategaonkar has constructed a principal right ideal domain of arbitrary (classical) Krull dimension [Jl]. In this note we will deal with the Noetherian case by showing that, if R is a Noetherian ring in which every right ideal can be generated by n elements, then R has classical Krull dimension at most one. If R is a polynomial extension of a subring then this is proved in [St2]. This is about as good a result as one can expect. Certainly one cannot replace classical Krull dimension by the Krull dimension of Rentschler and Gabriel, as any simple Noetherian ring of finite Krull dimension has a bounded number of generators for right ideals. Similarly, one cannot just require that every ideal has a bounded number of generators, since it i