Rings and semigroups with permutable zero products

We consider rings RR, not necessarily with 1, for which there is a nontrivial permutation σσ on nn letters such that x1⋯xn=0x1⋯xn=0 implies xσ(1)⋯xσ(n)=0xσ(1)⋯xσ(n)=0 for all x1,…,xn∈Rx1,…,xn∈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 Pn(R)Pn(R) consisting of those permutations σσ on nn letters for which the condition above is satisfied in RR. 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 Köthe conjecture.