On a generalization of right duo rings
A ring \(R\) is \textit{right duo} when every right ideal is a two-sided ideal. Equivalently, for every \(a\in R\) the containment \(Ra\subseteq aR\) holds. Similarly a ring is \textit{right quasi-duo} when every maximal right ideal is two-sided. These are important weakenings of commutativity that have found multiple applications in the literature.NEWLINENEWLINEStill following the literature, a ring \(R\) is called \textit{weakly right duo} if for every \(a\in R\) there exists some \(n\geq 1\) such that \(Ra^n\subseteq a^nR\). The paper under review generalizes this condition even further by introducing the following new definition: a ring \(R\) is \textit{right \(\pi\)-duo} if for every \(a\in R\) there exists some \(n\geq 1\) such that \(Ra^n\subseteq aR\). Clearly, weakly right duo rings are right \(\pi\)-duo. Much of the paper revolves around (re-)proving facts about weakly right duo rings for the more general class of right \(\pi\)-duo rings. For instance, the authors show that right \(\pi\)-duo rings are always abelian, right Ore, and right quasi-duo.NEWLINENEWLINEThe authors further show that the right \(\pi\)-duo property passes to factor rings, certain restricted upper-triangular matrix rings, and central localizations by non-zero-divisors, but not to Dorroh extensions or polynomial rings. There are many classes of rings where the abelian, right quasi-duo rings coincide with the weakly right duo rings, and thus also coincide with the right \(\pi\)-duo rings. A few examples and results are of this flavor, requiring the reader to forage through the literature for proofs.NEWLINENEWLINEIn this review, I have given a slightly different definition of right \(\pi\)-regularity than the authors. However, the conditions are equivalent, even for non-unital rings, by slightly modifying the proof of their Proposition 1.1.NEWLINENEWLINEFinally, I should mention there seems to be a hole in Example 1.2(1). First, since iterated Laurent series do not commute, the ring \(V_2\) is not a subring of \(K\). This problem can be fixed by instead working with the ring \(K:=F(x,y)\) of rational functions, and making other necessary changes. Second, the authors' degree argument is invalid in the skew power series ring \(Kt;\sigma\). Fortunately, the authors have discovered an alternate example of a right \(\pi\)-duo ring which is not weakly right duo, which they will make available to readers.
scientific article; zbMATH DE number 6594411