Let S = Σi=1n Rai be a finite normalizing extension of R and suppose that SM is a left S-module. Denote by crk(A) the dual Goldie dimension of the module A. We show that crk(RM) ≤ n · crk(SM) if either SM is artinian or the group homomorphism M → aiM given by x → aix is an isomorphism.
We identify some situations where mappings related to left centralizers, derivations and generalized -derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation , of a semiprime ring the mapping defined by for all is a free action. We also show that for a generalized -derivation of a semiprime ring with associated -derivation a dependent element of is also a dependent element of Furthermore, we prove that for a centralizer and...
An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra of the R-module , taking any a ∈ A to the right multiplication by a, is an isomorphism of algebras. In this case is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite...