PS-modules over Ore extensions and skew generalized power series rings (Q1751393)

Summary: A right \(R\)-module \(M_R\) is called a PS-module if its socle, \(\mathrm{Soc} \left(M_R\right)\), is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. Let \(R\) be an associative ring with identity, \(M_R\) a unitary right \(R\)-module, \(O = R \left[x; \alpha, \delta\right]\) Ore extension, \(M \left[x\right]_O\) a right \(O\)-module, \(\left(S, \leq\right)\) a strictly ordered additive monoid, \(\omega : S \rightarrow \mathrm{End} \left(R\right)\) a monoid homomorphism, \(A = \left[\left[R^{S, \leq}, \omega\right]\right]\) the skew generalized power series ring, and \(B_A = \left[\left[M^{S, \leq}\right]\right]_{\left[\left[R^{S, \leq}, \omega\right]\right]}\) the skew generalized power series module. Then, under some certain conditions, we prove the following: (1) If \(M_R\) is a right PS-module, then \(M \left[x\right]_O\) is a right PS-module. (2) If \(M_R\) is a right PS-module, then \(B_A\) is a right PS-module.