Automorphism-invariant non-singular rings and modules

A ring A is a right automorphism-invariant right non-singular ring if and only if A=S×T, where S a right self-injective regular ring and T is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. Over a ring A, each direct sum of automorphism-invariant non-singular right modules is an automorphism-invariant module if and only if the factor ring of the ring A with respect to its right Goldie radical is a semiprime right Goldie ring.