Duo, quasi-duo and ascending chain condition principal properties over skew PBW extensions (2018)
Ore extensions (introduced by Ore [33]) have been one of the most studied non-commutative structures in the last century. The skew polynomial rings (see Definition 1.1.1) are an Ore extensions that thanks to its similarity to the classical polynomial ring, currently seeks to “copy” the properties that already have the classic polynomial ring such as Noetherianity, some homological properties, the characterization of ideals, and others. In particular, and as the first topic of interest in this work, Marks [26] examined an extreme situation for skew polynomial rings: he asked when every left (or right) ideal is two-sided. It is important to say that for an ordinary polynomial ring, this case can no occur unless the ring is commutative (i.e., an ordinary polynomial ring is one-sided duo only if it is commutative), as it was proved by Hirano, Hong, Kim and Park ([13], Lemma 3). This result was extended in [25], Lemma 3.3, and precisely, Marks [26] obtained further generalizations of these results: he showed that if a non-commutative Ore extension R[x; σ, δ] which is a duo ring on one side exists, then it has to be right duo, σ must be non-injective and δ 6= 0 ([26], Theorems 1 and 2). He also obtained a list of necessary conditions to guarantee that the Ore extension R[x; σ, δ] to be right duo. Nevertheless, Matczuk [28] proved that non-commutative skew polynomial ring which are right duo rings do exist and that the necessary conditions obtained by Marks are not sufficient for the skew polynomial ring to be right duo. Actually, Matczuk’s paper is one of the most important articles about the characterization of non-commutative rings which are duo rings.