For a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-Armendariz ring, that is a generalization of both $pi$-Armendariz rings, and $(alpha,delta)$-compatible skew Armendariz rings. We first observe the basic properties of skew $pi$-Armendariz rings, and extend the class of skew $pi$-Armendariz rings through various ring extensions. We nex...

2 The Riordan skew algebra K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Skew algebra of formal power series under substitution . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Substitution of formal power series . . . . ....

We investigate the theory of skew (formal) power series introduced by Droste, Kuske [5, 6], if the basic semiring is a Conway semiring. This yields Kleene Theorems for skew power series, whose supports contain finite and infinite words. We then develop a theory of convergence in semirings of skew power series based on the discrete convergence. As an application this yields a Kleene Theorem prov...

We characterize skew polynomial rings and power series that are reduced right or left Archimedean.

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...