Pseudo-Differential Operator Rings with Armendariz-Like Condition

For a ring R with a derivation δ, we introduce and investigate a generalization of reduced rings which we call an Armendariz ring of pseudo-differential operator type (or simply 𝒟𝒪-Armendariz ring). Various classes of non-reduced 𝒟𝒪-Armendariz rings is provided and a number of properties of this generalization are established. Radicals of the pseudo-differential operator ring R((x ^?1, δ)), in terms of those of a 𝒟𝒪-Armendariz ring R, is established.