Rings with linearly ordered right annihilators
In this paper, the authors introduce the class of lineal rings, that is, rings \(R\) such that, for any subsets \(A,B\subseteq R\) we have \(\mathrm{ann}_{r}^{R}\left( A\right) \subseteq\mathrm{ann}_{r}^{R}\left( B\right) \) or \(\mathrm{ann}_{r}^{R}\left( B\right) \subseteq \mathrm{ann}_{r}^{R}\left( A\right) \), where \(\mathrm{ann}_{r}^{R}\left( A\right) \) (resp., \(\mathrm{ann}_{r}^{R}\left( B\right) \)) denotes the right annihilator of \(A\) in \(R\)(resp., the right annihilator of \(B\) in \(R\) ). \ They give many examples of lineal rings and establish some of their properties. For example, they show that there is no need to distinguish between ``right lineal'' and ``left lineal'' and the characteristic of a lineal ring is either \(0\) or a prime power. The authors study zero-divisors of lineal rings and show how linear rings are related to some important classes of rings such as reduced rings, domains, right strongly prime rings, left strongly prime rings, Armendariz rings (that is, rings \(R\) \ such that \(\left( \sum_{i}a_{i}x^{i}\right) \left( \sum_{j}b_{j}x^{j}\right) =0\) in the polynomial ring \(R\left[ x\right] \) implies that \(a_{j}b_{j}=0\) for all \(i,j\)), right distributive rings (that is, rings whose right ideals lattices are distributive) or right Bezout rings (that is, rings whose finitely generated right ideals are principal), for example. In particular, they show that in any lineal ring the right zero-divisors form a right ideal, and that this property describes linear rings among distributive rings as well as among Bezout rings. The authors prove that a polynomial ring \(R\left[ x\right] \) is lineal if and only if the ring \(R\) \ is lineal and give an example of a lineal ring that is not Armendariz. They show, however, that if a lineal ring \(R\) contains infinitely many central elements whose diferences are regular, then \(R\) is Armendariz. The authors carefully examine nilpotent elements and nilradicals of linear rings. For example, they show that for any lineal ring \(R\), the set of nilpotent elements of \(R\) is a (nonunital) subring of \(R\) which is equal to the sum of its own nilpotent ideals so that, as in the case of commutative rings, one can speak of the nilradical of a lineal ring. Moreover, the authors prove that several standard nilradicals coincide in the class of lineal rings and that the famous Kothe Conjecture (which states that a ring with no nonzero nil ideals has no nil one-sided ideals) has a positive answer in the class of lineal rings.
scientific article; zbMATH DE number 6673806
scientific article