Abstract: | It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent.
If, moreover, the R-module M and the left idealR(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a
semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings.
Moreover, first, a right Artinian ring R such that rad(R)2=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful
Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely
generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension.
Supported by RFFR grant No. 26-93-011-1544.
Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997. |