Journal of Pure and Applied Algebra 206 (2006) 355 – 369
Rings and semigroups with permutable zero
, Andrzej Kisielewicz
Université Blaise Pascal, Laboratoire de Mathématiques, 63177 Aubière Cedex, France
University of Wrocław, Institute of Mathematics, pl. Grunwaldzki 2, 50-384 Wrocław, Poland
Received 27 July 2003; received in revised form 20 July 2005
Available online 30 September 2005
Communicated by J. Rhodes
We consider rings R, not necessarily with 1, for which there is a nontrivial permutation on n letters
such that x
= 0 implies x
= 0 for all x
∈ R. We prove that this condition
alone implies very strong permutability conditions for zero products with sufﬁciently many factors. To
this end we study the inﬁnite sequences of permutation groups P
(R) consisting of those permutations
on n letters for which the condition above is satisﬁed in R. We give the full characterization of such
sequences both for rings and for semigroups with 0. This enables us to generalize some recent results
by Cohn on reversible rings and by Lambek, Anderson and Camillo on rings and semigroups whose
zero products commute. In particular, we prove that rings with permutable zero products satisfy the
© 2005 Elsevier B.V. All rights reserved.
MSC: 16U80; 16P10; 16S99; 20B30; 20B35
In , P.M. Cohn has introduced reversible rings as those rings with 1 in which ab = 0
implies ba = 0. He has observed that this condition helps to simplify other ring conditions,
This research was done while the second author was visiting Université Blaise Pascal in Clermont-Ferrand.
Supported in part by Polish KBN Grant P03A 01124.
E-mail addresses: Marin.Gutan@math.univ-bpclermont.fr (M. Gutan), firstname.lastname@example.org
0022-4049/$ - see front matter © 2005 Elsevier B.V. All rights reserved.