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Local finiteness in the sense of Shirshov. (Russian) Zbl 0544.17001
Let \(\Phi\) be an associative commutative ring with identity, Z an ideal of \(\Phi\). A finitely generated \(\Phi\)-algebra A is said to be finite over Z in Shirshov’s sense if there exist elements \(a_ 1,...,a_ n\in A\) and a natural number m such that \(A^ m\subseteq \sum^{k}_{i=1}Za_ i\). An algebra B is said to be locally finite over Z if any finitely generated \(\Phi\)-subalgebra of B is finite over Z. This notion which was first introduced by A. I. Shirshov generalizes the known notions of local nilpotency and local finite-dimensionality which one obtains by choosing \(Z=0\) or \(Z=\Phi\) correspondingly. It was proved in [K. A. Zhevlakov and the reviewer, Algebra Logika 12, 41-73 (1973; Zbl 0289.17001)] that the property of local finiteness in Shirshov’s sense is a radical one in the varieties of alternative, Jordan, (-1,1)-algebras and in some other varieties.
The purpose of the author is to construct some universal locally finite algebra such that any other algebras of the given dimension are its homomorphs. To be more exact, let \(\kappa\) be a fixed cardinal number, Z be an ideal of \(\Phi\) with card \(Z\leq \kappa\), \({\mathcal K}\) be one of the varieties of associative, alternative, Jordan or (-1,1)-algebras over \(\Phi\). Then there exists an algebra \(R\in {\mathcal K}\) locally finite over Z in Shirshov’s sense with dim \(R\leq \kappa\) such that any other algebra \(A\in {\mathcal K}\) locally finite over Z with dim \(A\leq \kappa\) is a homomorph of R. Earlier Yu. M. Ryabukhin has constructed locally nilpotent associative algebras with the same properties [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 1203-1223 (1976; Zbl 0342.08008)].
Reviewer: I.P.Shestakov
MSC:
17A60 Structure theory for nonassociative algebras
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16Rxx Rings with polynomial identity
17C65 Jordan structures on Banach spaces and algebras
17D05 Alternative rings
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