Abraham A. Klein

The sum of nil one-sided ideals of bounded index of a ring

The sumN(R) of nil one-sided ideals of bounded index of a ringR is shown to coincide with the set of all strongly nilpotent elements ofR of bounded index. The known result thatN(R) is contained in the prime radical is highly improved and it is shownN(R) is contained inN2(R). It is proved that the sum of a finite number of nil left ideals of bounded index has bounded index.

Bounds for indices of nilpotency and nility

Abstract. The known bound for the index of nilpotency of a finitely generated nil ring of bounded index is improved and the new bound is applied to obtain other bounds for indices of nilpotency and nility.

Rings with finitely many nilpotent elements

It is well-known that a ring with no nonzero nilpotent elements - a so-called reduced ring - is a subdirect product of domains. Moreover, as we have recently shown [2], a prime ring with only finitely many nilpotent elements is either a domain or is finite. In view of these results, it is natural to ask what can be said in general about rings with only finitefy many nilpotent elements. A crucial property of such rings is that they contain no infinite zero subrings, hence we are led to consider rings with this property also. Our principal result is that for any ring R with only finitely many nilpotent elements is a direct sum of a reduced ring and a finite ring, where p(R) denotes the prime radical of R. One consequence is a finiteness theorem for periodic rings; another is the rather surprising result that every ring with infinitely many nilpotent elements has an infinite zero subring.

Some commutativity results for rings

It is proved that certain rings satisfying generalized-commutator constraints of the form [ x m , y n , y n , …, y n ] = 0 must have nil commutator ideal.

A COMMUTATIVITY-OR-FINITENESS CONDITION FOR RINGS

We show that a ring with only finitely many noncentral subrings must be either commutative or finite.

COMBINATORIAL COMMUTATIVITY AND FINITENESS CONDITIONS FOR RINGS

Bk-groups are a natural generalization of abelian groups, and it is reasonable to ask under what conditions a Bk-group must be abelian. In [6] Freiman gave a complete characterization of B2-groups: they are either abelian or of form Q6E, where Q is the quaternion group of order 8 and E is an elementary abelian 2-group. A decade later Brailovsky [4] showed that for k> 2, a Bk-group is either abelian, or finite with order bounded by a number M1⁄4M(k). The property ( )k is equally natural for rings, and we call a ring R a Bk-ring if it satisfies ( )k. Some years ago we proved that every B2-ring with

The codimensions of aPi-algebra

We simplify the numerical calculations given in a previous paper by Regev and obtain a much better estimation for the sequence of codimensions of aPI-algebra.

Noncommutativity and noncentral zero divisors

Let R be a ring, Z its center, and D the set of zero divisors. For finite noncommutative rings, it is known that D\Z≠∅. We investigate the size of |D\Z| in this case and, also, in the case of infinite noncommutative rings with D\Z≠∅.

A note about two properties of matrix rings

We show that there exists for eachm≧2 a (non-commutative) integral domainR with a nilpotent matrixC ∈Rm whose order of nilpotency is greater thanm, and anyA ∈Rm with a right (or a left) inverse is invertible.

On Commutativity and Centrality in Infinite Rings

We show that if R is an infinite ring such that XY ∩ YX ≠ ∅ for all infinite subsets X and Y, then R is commutative. We also prove that in an infinite ring R, an element a ∈ R is central if and only if aX ∩ Xa ≠ ∅ for all infinite subsets X.