Invariant rings and quasiaffine quotients

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. In this paper we will show that nevertheless it is always isomorphic to the ring of functions on a quasi-affine variety. Mathematics Subject Classification (2000): 13A50, 14R20, 14L30 Received: 12 April 2002 / Published online: 24 February 2003     

Prime rings with PI rings of constants

It is shown that if the ring of constants of a restricted differential lie algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI) then the original prime ring has a generalized polynomial identity (GPI). If additionally the ring of constants is semiprime then the original ring is PI. The case of a non-quasi-Frobenius inner part is also considered.     

A note on complete rings of quotients and McCoy rings

If a (commutative unital) ring $A$ is reduced and coincides with its total quotient ring, then $A$ satisfies Property A (that is, $A$ is a McCoy ring) if and only if the inclusion of $A$ in its complete ring of quotients $C(A)$ is a survival extension. The ??if?? assertion fails if one deletes the hypothesis that $A$ is reduced. This is shown by using the idealization construction to construct a suitable ring $A$ and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring $A$ is von Neumann regular if and only if $A$ is a reduced McCoy ring that coincides with its total quotient ring such that $A \subseteq C(A)$ satisfies GD.     

The rings of quotients of free algebras

It is proved that a symmetric Utumi ring of quotients, U, of a free associative (noncommutative) algebra F(X) with unity coincides with the algebra itself, U=F(X). From this, we obtain a similar statement concerning a symmetric Martindale ring of quotients, Q(F(X))=F(X), which is well known. In addition, it is shown that a left Martindale ring of quotients, F(X)_F, of a free algebra is a prime algebra and, moreover, every homogeneous element in a free algebra has the right inverse in F(X)_F but does not have the left one (unless, of course, r belongs to an underlying field). Since a left Utumi ring of quotients and a left Martindale ring of quotients for a free algebra both appear prime, an interesting question arises as to whether or not they coincide. Supported by RFFR grant No. 95-01-01356 and by ISF grant RPS000-RPS300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 655–662, November–December, 1996.