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
Full Text:

References:

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