(untitled) (European Journal of Pure and Applied Mathematics, 2023)
Let R be a ring and M be a monoid with a twisting map f : M × M → U(R) and an action map ω : M → Aut(R). The objective of our work is to extend the reflexive properties of rings by focusing on the crossed product R * M over R. In order to achieve this, we introduce and examine the concept of strongly CM-reflexive rings. Although a monoid M and any ring R with an idempotent are not strongly CM-reflexive in general, we prove that R is strongly CM-reflexive under some additional conditions. Moreover, we prove that if R is a left p.q.-Baer (semiprime, left APP-ring, respectively), then R is strongly CM-reflexive. Additionally, for a right Ore ring R with a classical right quotient ring Q, we prove R is strongly CM-reflexive if and only if Q is strongly CM-reflexive. Finally, we discuss some relevant results on crossed products.