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Kostia’s contribution to radical theory and related topics. (English) Zbl 1130.16015

Chebotar, Mikhail (ed.) et al., Rings and nearrings. Proceedings of the international conference of algebra in memory of Kostia Beidar, Tainan, Taiwan, March 6–12, 2005. Berlin: Walter de Gruyter (ISBN 978-3-11-019952-9/hbk). 121-157 (2007).
This is a very nice survey article on Kostia Beidar’s rich contributions to radical theory and related topics as well as on the considerable impact his findings made on other researchers. Each section of the paper is devoted to a certain topic, including general radical theory; concrete radicals and structure theory; rings with involution and nonassociative rings.
The general radical theory section includes a discussion of the famous Sulinski-Anderson-Divinsky problem which asks whether there are examples of classes of rings for which the lower radical construction terminates in precisely \(3\), \(4,\dots,\omega\) steps. Beidar’s solution to this problem is presented and several related results are also given. This section also contains Beidar’s famous characterization of special radicals and its important consequences; Beidar’s positive solution to the question posed by Gardner which asks whether there exist disjoint special classes \(\rho_1\) and \(\rho_2\) which generate the same upper radical and Beidar’s and Salavova’s positive solution to the question of Szász which asks whether there are infinitely many non-special supernilpotent radical classes \(\gamma_1,\gamma_2,\dots\) such that \(\mathcal S\gamma_n\cap\mathcal S\gamma_m=\{0\}\) for \(m\neq n\), where \(\mathcal S\gamma\) denotes the semisimple class of a radical \(\gamma\). This section also contains Beidar’s brilliant results concerning dependence and independence among radicals involving one-sided ideals of associative rings and his explicit description of radical classes with semisimple essential cover. Beidar’s significant contribution to the knowledge on lattices of radicals such as description of atoms of lattices of radicals and description of complemented radicals in the lattice of all radicals is shown in this section, too.
The second section of the paper contains Beidar’s famous results concerning the Jacobson radical of finitely generated algebras over countable fields as well as his brilliant results related to Köthe’s nil ideal problem, in particular, those concerning some radicals of polynomial rings. Moreover, this section contains the main results on left (right) extensions of reduced rings and domains obtained by Beidar, Ke, Fong and Puczyłowski as well as Beidar’s and Wiegandt’s results concerning radicals induced by the total of rings. Several important results of Beidar and Wiegandt which concern involution rings with dcc on *-biideals and involution rings with acc on *-biideals are given in the third section of the paper. The involutive version of the density theorem for rings as well as Beidar’s description of *-primitive rings are also included in this section.
The last section of the paper contains Beidar’s sufficient conditions on universal classes of nonassociative rings for a well-behaved radical theory as well as Beidar’s and Wiegandt’s splitting of the torsion radical theorems for alternative and Jordan rings.
The paper is very well organized and also has an extensive list of references.
For the entire collection see [Zbl 1113.16001].

MSC:

16N80 General radicals and associative rings
01A70 Biographies, obituaries, personalia, bibliographies
16N20 Jacobson radical, quasimultiplication
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
17A60 Structure theory for nonassociative algebras
17A65 Radical theory (nonassociative rings and algebras)
17D05 Alternative rings

Biographic References:

Beidar, Kostia
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