  Let R be a ring (not necessarily with 1) and G be a finite group of
automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not
p-torsion free, is called the set of bad primes. When the ring is |G|-torsion
free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are
closely connected. The aim of the paper is to show that this is also true when
B(R, G) is not empty set under natural conditions on bad primes. In particular,
it is shown that the Jacobson radical (resp., the prime radical) of the ring
R^G is equal to the intersection of the Jacobson radical (resp., the prime
radical) of R with R^G; if the ring R is semiprime then so is R^G; if the trace
of the ring R is nilpotent then the ring itself is nilpotent; if R is a
semiprime ring then R is left Goldie iff the ring R^G is so, and in this case,
the ring of G-invariants of the left quotient ring of R is isomorphic to the
left quotient ring of R^G and \udim (R^G)\leq \udim (R)\leq |G| \udim (R^G).
