On equational theories of varieties of anticommutative rings

The existence of a finitely based variety of anticommutative rings (in the sense of the identityx ²=0) with unsolvable equational theory is proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 230–245, February, 1999.     

On the decidability of equational theories of varieties of rings

A result illustrating the complexity of describing the varieties of rings with undecidable equational theory is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 873–881, June, 1998.The author wishes to thank Yu. M. Vazhenin for his supervision of this research.     

Essentially nilpotent rings

In this note we introduce a class of nil rings (called essentially nilpotent) which properly contains the class of nilpotent rings. A nil ring is said to be essentially nilpotent if it contains an essential right ideal which is nilpotent. Various properties of essentially nilpotent rings are investigated. A nil ring is essentially nilpotent if and only if it contains an essential right ideal which is leftT-nilpotent.     

Equational theories for the varieties of metabelian and commutative rings

Examples of finitely based varieties of metabelian and commutative rings for which equational theories are undecidable are constructed; critical theories are pointed out. Translated from Algebra i Logika, Vol. 34, No. 3, pp. 347-361, May-June, 1995.     

On differential polynomial rings over locally nilpotent rings

Let δ be a derivation of a locally nilpotent ring R. Then the differential polynomial ring R[X; δ] cannot be mapped onto a ring with a non-zero idempotent. This answers a recent question by Greenfeld, Smoktunowicz and Ziembowski.     

On skew polynomial rings over locally nilpotent rings

We will show that skew polynomial rings in several variables over locally nilpotent rings cannot contain nonzero idempotent elements. We will also prove that such rings are Brown–McCoy radical.     

On group rings of nilpotent groups

It is shown that ifG is a non-abelian torsion free nilpotent group andF is a field, then the classical skew field of fractionsF(G) of the group ring,F[G] contains a noncommutative free subalgebra. The author is supported by NSF Grant No. MCS-8201115.