I.N Herstein

Multiplicative commutators in division rings

In this paper we study division rings in which the multiplicative commutators are periodic or periodic relative to the center.

Invariant subrings of a certain kind

The theorem which we shall prove here states that if a subring of a prime ring is invariant with respect to a certain class of automorphisms then a dichotomy of the Brauer-Cartan-Hua type exists.

The intersection of the powers of the radical in Noetherian P.I. rings

LetR be a ring and J its radical. DefineJ1=∩Jn, J2=∩J1n,…,… Jk=∩Jk−1n.... It is shown that in a ringR satisfying a polynomial identity and the ascending chain condition on ideals,Jk=0 for some appropriatek.

Rings with periodic symmetric or skew elements

Recently there have been extensions to the case of rings with involution of the classical theorem of Jacobson [6, l] w ic h h asserts that if every element zc in a ring R is periodic, in the sense that &jc) = a for some n(x) > 1, then R must be commutative. Now, as is easy to see, if F is a subfield of the algebraic closure of a finite field of characteristic not 2, then the ring R of all 2 x 2 matrices over F relative to the symplectic involution defined via (

A Lie algebra variation on a theorem of Wedderburn

)* = (5, -E) satisfies the relation P”) = S, n(s) > 1, for every s E R such that s* = S. With F and R as above but using the involution given by the transpose we see that KncB) = K, n(k) > 1, for every K E R such that k* = --K. Thus, imposing the Jacobson condition of periodicity on all the symmetric elements of a ring with involution, or on all the skew elements of such a ring, is clearly not sufficient to force the commutativity of that ring. However, there is a great deal that one can say such rings. For division rings with involution Herstein and Montgomery [4] showed that when the symmetric elements, or the skew elements, satisfied Jacobson’s condition the division ring was indeed commutative. Montgomery [9, IO] then showed that the structure of a semiprime ring with periodic symmetric elements is that of a subdirect sum of fields (algebraic over finite fields) and of 2 x 2 matrix rings over such fields. The case of periodic skew elements, however, was left open. In this paper we show that Montgomery’s results (except for some statements of finiteness of fields) also hold for the structure of rings whose skew elements are periodic. In doing this we give, at the same time and via the same proof, a new proof of Montgomery’s theorems. While her proof made use of some results in Jordan algebras due to Osborn, we shall have no need to use or cite these results.

Center-like elements in prime rings

THEOREM. Let A be a finite dimensional associative algebra over a field F. Suppose that A has a basis over F consisting of nilpotent elements. Then A itself must be nilpotent. Given a finite dimensional Lie algebra L which is a Lie subalgebra of an associative algebra A (possibly infinite dimensional), we can ask what results from supposing that L has a basis over F consisting of elements that are nilpotent in the associative algebra. We cannot conclude that L is nilpotent and finite dimensional. In fact, this is rarely the case. For example,