Higher commutators, ideals and cardinality

Abstract

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [ [R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2 e " d T and there is a central ideal of R of index at most 2 c " d T in R.In this paper we investigate the relative cardinalities of certain subsets of a ring -R, with the specific goal of generalising work of Hirano [4] who showed the equivalence in a semiprime ring R of the finiteness of the commutator ideal, the additive index of the centre of R, and the existence of a central ideal of finite index in R. Our aim is to generalise these results to the set of commutators, to higher commutators, and to arbitrary cardinalities, and also to consider the relation between the cardinality of higher commutators and the ideals they generate. A number of papers in the literature relate the cardinality of a ring to that of special subsets. To mention a few of these, in [7], Koh shows that a ring is finite if it has finitely many left zero divisors, and at least two, Hirano extended this to finitely many two-sided zero divisors [3], and recently we considered the relation between the cardinality of a ring and that of the ideal generated by various subsets of nilpotent elements [8]. We have also studied the relation between the cardinality of a ring and that of the subring generated by the set of integral elements or the set of symmetric elements in a ring with involution [9].Throughout the paper, R will denote an associative ring, not necessarily with unit, and Z(R) = Z will be the centre of R. For nonempty subsets A, BCR, L(A, B) = { [a, b] = ab-ba | a £ A and 6 6 B} is the set of (Lie) commutators of A with B, [A, B] is the additive subgroup of R generated by L (A, B), (A) is the subring generated by A, and (.A) is the ideal of R generated by A. We postpone a discussion of higher commutators until later. Our first results will generalise those in [4] by considering L(R, R) rather than the commutator ideal ( [R, R]), and by considering

Finite Higher Commutators in Associative Rings

“…We cannot prove a complete generalisation of [8], but we are able to prove that if T is a finite higher commutator that generates an infinite ideal, and if the cardinality of such a T is minimal, then T must be a central subring with trivial multiplication. If there is an example of a finite T in a ring R generating an infinite ideal, then the c 2013 Australian Mathematical Publishing Association Inc. 0004-9727/2013 $16.00 direct sum of R with suitable matrix rings over finite fields would have a finite higher commutator generating an infinite ideal but neither central nor nilpotent.…”

“…In [8] we showed that for any finite higher commutator W in a ring R, the higher commutator [W, R] generates a finite ideal, denoted by ([W, R]). Thus the interest here is in higher commutators not of this form, although there seems to be no advantage in assuming this to start with.…”

“…These led to [9] and [10], which also considered subsets with infinite cardinality. Motivated by these sorts of results, we showed in [8] that a finite higher commutator in a semiprime ring must generate a finite ideal in that ring. A simple commutator in a ring R is any xy − yx = [x, y] for x, y ∈ R. A higher commutator, defined below, can be thought of as an additive subgroup of (R, +) generated by a fixed succession of commutators.…”

“…For general rings it is difficult to characterise these sets in other ways, to understand the relations between them, or to determine the ideals they generate. The purpose of this paper is to generalise [8]: must any higher commutator in R that has only finitely many elements generate a finite ideal in R? We know no example of an infinite ring having a finite higher commutator that generates an infinite ideal.…”

“…We also prove a result showing that any finite higher commutator that generates an infinite ideal must contain nonzero nilpotent elements. Our approach here does not extend to the case of higher commutators of infinite cardinality, which was the case in [8].…”