Let R be a nil ring with p R = 0 for some prime number p. We show that the polynomial ring R[x,y] in two commuting indeterminates x, y over R cannot be homomorphically mapped onto a ring with identity. This extends, in finite characteristic case, a result obtained by Smoktunowicz [] and gives a new approximation, in that case, of a positive solution of Köthe's problem.
Let N be a homomorphically closed class of associative rings. Put N1 = N 1 = N and for ordinals α ≥ 2, define Nα, (Nα) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in Nβ for some β < α. In this way we obtain a chain {Nα} ({Nα}), the union of which is equal to the lower radical class lN (lower left strong radical class lsN ) determined by N . The chain {Nα} is called Kurosh’s chain of N . In [11] Sulinski, Anderson and Divinsky studied the Kurosh’s chain in the universal class of associative rings. They described the classes Nα in terms of accessible subrings and proved that the chain stabilizes at the first limit ordinal ω. They asked whether, for each ordinal α ≤ ω, there exists a class N such that lN = Nα 6= Nβ for β < α. The question turned out to be very interesting and challenging. In [7], Heinicke answered it in the positive for α = ω. However, the problem for positive integers resisted effords of many authors for a long time. Various studies, except some general results, effected in solving the problem for small integers and determining when the chain stabilizes for some specific classes N . The general problem was finally solved in the positive by Beidar in [3]. After that, several new examples were found (see [1], [4], [9], [1! 0], [12]). All of them were not easy to handle and required quite complicated arguments. Developing some ideas of the mentioned papers (mainly those of [10]), we construct new general examples. The generality allows us to avoid particular calculations which, we hope, makes the arguments clearer and more visible. It also allows us to construct radicals which satisfy some extra properties. References. [1] R. R. Andruszkiewicz, E. R. Puczy lowski, Kurosh’s chains of associative rings, Glasgow Math. J. 32 (1990), 67–69. [2] E.R. Armendariz, W. G. Leavitt, The hereditary property in the lower radical construction, Canad. J. Math. 20 (1968), 474–476. [3] K. I. Beidar, A chain of Kurosh may have an arbitrary finite length, Czech. Math. J. 32 (1982), 418–422. [4] K. I Beidar, Semisimple classes and the lower radical, Mat. Issled. 105 (1988), 8–12 (in Russian). [5] N. Bourbaki, Algebre Commutative, Hermann, Paris, 1961–1965. [6] G. Ehrlich, Filial rings, Portugaliae. Math. 42 (1983/1984), 185–194. [7] A. Heinicke, A note on lower radical constructions for associative rings, Canad. Math. Bull. 11 (1968), 23–30. [8] A. G. Kurosh, Radicals of rings and algebras, Mat. Sbornik. 33 (1953), 13–26. [9] S. X. Liu, Y. L. Luo, A. P. Tang, J. Xiao, J. Y. Guo, Some results on modules and rings, Publ. Soc. Math. Belg. Ser, B, 39 (1987), 181–193.
We discuss several problems on the structure of nil rings from the linear algebra point of view. Among others, a number of questions and results are presented concerning algebras of infinite matrices over nil algebras, and nil algebras of infinite matrices over fields, which are related to the famous Koethe's problem. Some questions on radicals of tensor products of algebras related to Koethe's problem are also discussed.
Abstract We survey some old and recent results concerning the Goldie dimension of modules and modular lattices and its properties which are counterparts of properties of the dimension of linear spaces.
A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a strongly Z-graded ring is left quasi-duo if and only if it is right quasi-duo. This gives a partial answer to a problem posed by Dugas and Lam in [1].
A commutative ring R is called 2-absorbing (Badawi in Bull. Aust. Math. Soc. 75:417–429, 2007) if for arbitrary elements a,b,c∈R, abc=0 if and only if ab=0 or bc=0 or ac=0. In this paper we study this concept in a more general framework of commutative (multiplicative) semigroups with 0. The results obtained apply to many ring theoretic situations and make it possible to describe similarities and differences among some variants of the notion. We pay a particular attention to graded rings. We also show that a conjecture from (Anderson and Badawi in Commun. Algebra 39:1646–1672, 2011) concerning n-absorbing rings holds for rings with torsion-free additive groups.
It is shown that methods of matroid theory can be applied in studies of some dimensions of modular lattices and modules. In particular one can obtain some fundamental properties of the Goldie and Kuroš-Ore dimensions of modular lattices.
Let N be a homomorphically closed class of associative rings. Put N 1 = N l = N and, for ordinals a ≥ 2, define N α ( N α ) to be the class of all associative rings R such that every non-zero homomorphic image of R contains a non-zero ideal (left ideal) in N β for some β<α. In this way we obtain a chain { N α } ({ N α }), the union of which is equal to the lower radical class IN (lower left strong radical class IsN ) determined by N . The chain { N α } is called Kurosh's chain of N . Suliński, Anderson and Divinsky proved [7] that . Heinicke [3] constructed an example of N for which lN ≠ N k for k = 1, 2,. … In [1] Beidar solved the main problem in the area showing that for every natural number n ≥ 1 there exists a class N such that IN = N n +l ≠ N n . Some results concerning the termination of the chain { N α } were obtained in [2,4]. In this paper we present some classes N with N α = N α for all α Using this and Beidar's example we prove that for every natural number n ≥ 1 there exists an N such that N α = N α for all α and N n ≠ N n+i = N n+2 . This in particular answers Question 6 of [4].
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