| Abstract: | 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. |
