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On subrings of free rings. (English. Russian original) Zbl 0723.17001

Sib. Math. J. 30, No. 6, 903-914 (1989); translation from Sib. Mat. Zh. 30, No. 6(178), 87-97 (1989).
Let k be a commutative associative domain with unity, and M one of the following varieties: (i) variety of all k-algebras; (ii) variety of all (anti) commutative algebras; (iii) variety of all Lie algebras; (iv) variety of all Lie p-algebras, \(p=char k.\)
Let F be a free M-algebra. A subalgebra B in F is called isolated if ax\(\in B\), where \(a\in k\setminus 0\), implies \(x\in B\). A subalgebra B in F is diagonal if there exists a free generating set X in F and elements \(a_ x\in k\setminus 0\) such that B is generated by \(a_ xx\), \(x\in X\). If C is a subalgebra in F and k a principal ideal ring, then there exists a diagonal subalgebra D in F such that \(D\subseteq C\subseteq F\). For some special domains k it is proved the existence of algorithms deciding whether (i) a system of equations over F is soluble; (ii) for two elements f,g\(\in F\) there exists an endomorphism a in F such that \(a(f)=g\).

MSC:

17A30 Nonassociative algebras satisfying other identities
17B01 Identities, free Lie (super)algebras
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References:

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