We introduce and study comodules over cowreath defined over a given coring. If our coring
arises from an entwining structure, we then give a procedure to construct a cowreath from a
given cowreath over the factor coalgebra. We also include the dual notions, that is, wreaths
over ring extension and their modules. In particular, we show that the study of twisted
algebras and twisted bimodules already introduced in [A. Čap, H. Schichl and J. Vanžura,
On twisted tensor products of algebras, Commun. Algebra 23 (1995) 4701–4735] has its …

A basic theory of cowreath or extended distributive laws in the bicategory of unital
bimodules, is deciphered. Precisely, we give in terms of tensor product over a scalar base
ring, a simplest and equivalent definition for cowreath over coring and for comodule over
cowreath. An adjunction connecting the category of comodules over the factor coring and
the category of comodules over the coring arising from the cowreath products is also given.
The dual notions ie wreaths over rings extension and their modules are included.