Modules and comodules for corings

A coring $C$ over a ring $A$ is an $(A,A)$-bimodule with a comultiplication $\Delta\colon C\to C\otimes_AC$ and a counit $\varepsilon\colon C\to A$, both being left and right $A$-linear mappings satisfying additional conditions. The dual spaces $C^*=\mathrm{Hom}_A(C,A)$ and ${}^*C={}_A\mathrm{Hom}(C,A)$ allow the ring structure and the right (left) comodules over $C$ can be considered as left (right) modules over ${}^*C$ (respectively, $C^*$). In fact, under weak restrictions on the $A$-module properties of $C$, the category of right $C$-comodules can be identified with the subcategory $\sigma[{}_{^*C}C]$ of ${}^*C$-Mod, i.e., the category subgenerated by the left ${}^*C$-module $C$. This point of view allows one to apply results from module theory to the investigation of coalgebras and comodules.