Reversible and symmetric rings.
A ring is called right (resp. left) duo if every right (resp. left) ideal is two-sided. A ring \(R\) is said to be reversible if \(ab=0\) is equivalent to \(ba=0\) for all \(a,b\in R\). A stronger condition than reversible is symmetric. A ring \(R\) is called symmetric if \(abc=0\) if and only if \(acb=0\) for all \(a,b,c\in R\). In this paper, precise relationships among these three conditions are studied. An interesting counterexample showing that the reversible condition does not carry over to the Dorroh extension is established. Also another interesting counterexample of a (right and left) duo reversible ring which is not a symmetric ring is constructed.
scientific article; zbMATH DE number 1864812