Locally Compact Topologically Nil and Monocompact PI-rings

In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely generated PI-algebra. We will use also the locally projectively nilpotent radical of a locally compact ring constructed in [5]. For a discrete Φ-algebra R the locally nilpotent radical in the class of Φ-algebras coincides with the locally nilpotent radical of the ring R (considered as a Z-algebra). We give an example which shows that for locally compact Φ-algebras the locally projectively nilpotent radical does not always exist. K. I. Beidar posed the following question: Let R be a simple nil ring. Does R admit a non-discrete locally compact ring topology? We proved in [7] that if R is a simple nil ring then R doesn’t admit a locally compact ring topology relative to which it can be represented as a union of a familiy of cardinality < c compact subsets. In particular, there are no second countable locally compact ring topologies on R. We will give in this paper other partial answers to the question of K. I. Beidar. In this context, let us mention that the longstanding problem of the existence of a simple nil ring has been solved recently affirmatively by A. Smoktunowicz [3].