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On positive and critical theories of some classes of rings. (English. Russian original) Zbl 0951.17001

Sib. Math. J. 41, No. 2, 224-228 (2000); translation from Sib. Mat. Zh. 41, No. 2, 278-283 (2000).
Let \(\mathfrak X\) be some variety of rings defined by polyhomogeneous identities. The author proves that the positive theories of the following varieties coincide: (i) \(\mathfrak X\); (ii) the free ring in \(\mathfrak X\) of countable rank \(F\mathfrak X\); and (iii) the class \(\mathfrak X\cap F\) of all finite rings in \(\mathfrak X\).
Let \(\mathfrak N\) be an arbitrary variety of semigroups, groups, rings, or algebras over a field. Let \(F\mathfrak N\) be a free object in the variety \(\mathfrak N\) of countable rank and let \(\mathfrak N\cap F\) be the class of all free objects of the variety. The authors show that the positive theory of the algebraic system \(F\mathfrak N\) and that of the class \(\mathfrak N\cap F\) coincide.
As a corollary, the authors give a negative answer to Bokut’s question on decidability of the word problem for the class of finite Jordan rings.

MSC:

17A01 General theory of nonassociative rings and algebras
17A30 Nonassociative algebras satisfying other identities
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References:

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