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On the \(R\)-center of Mal’tsev algebras. (Russian) Zbl 0751.17030

Let \(A\) be a Mal’tsev \(\Phi\)-algebra, where \(\Phi\) is a commutative- associative ring with 1/6, \(R(A)\) be the algebra of right multiplications of \(A\), \(\text{Ann }R(A)\) be the annihilator of \(R(A)\), and \(J(A)\) be the ideal of \(A\) generated by all Jacobians. The center of \(R(A)\), denoted by \(Z_ R(A)\), is called the \(R\)-center of \(A\). Among the results established here are the following:
(1) If \(A\) is a free algebra from \(k\geq 5\) generators, then \(\text{Ann }R(A)\neq 0\). (2) If \(A\) is a free algebra from \(k\geq 3\) generators, then \(Z_ R(A)\neq \text{Ann }R(A)\). (3) There is exhibited an identity of degree 6 which is satisfied by \(A^ 2\) but not by \(A\). (4) \(J(A^ 2)\) can be isomorphically imbedded in the commutator algebra of some alternative algebra. (5) \(J(A^ 2)\) is Shirshov locally finite over the \(R\)-center of \(A^ 2\). (6) \(J(A)^ 2\) is Shirshov locally finite over the \(R\)-center of \(J(A)\).

MSC:

17D10 Mal’tsev rings and algebras
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