Monoidal ring and coring structures obtained from wreaths and cowreaths

Let $A$ be an algebra in a monoidal category $\text{Cc}$, and let $X$ be an object in $\text{Cc}$. We study $A$-(co)ring structures on the left $A$-module $A\text{ot}X$. These correspond to (co)algebra structures in $EM(\text{Cc})(A)$, the Eilenberg-Moore category associated to $\text{Cc}$ and $A$. The ring structures are in bijective correspondence to wreaths in $\text{Cc}$, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in $\text{Cc}$, and their category of corepresentations is the category of generalized entwined modules. We present several examples coming from (co)actions of Hopf algebras and their generalizations. Various notions of smash products that have appeared in the literature appear as special cases of our construction.