Semisimple corings

Abstract

While semisimple artinian rings and semisimple coalgebras over a field can be described in terms of matrices (either matrix ring over division rings or comatrix coalgebras over the ground field), semisimple corings seem to have a more intrincated structure in general. It turns out that some well-known properties of semisimple rings or coalgebras, which are immediately deduced from the aforementioned structure, are not evident over a (left) semi-simple coring. For instance, it is not evident that the notion of semi-simple coring is left-right symmetric. To be precise, if every left comodule decomposes a a direct sum of simple comodules, do the right comodules have such a decomposition? In other words, is every left semi-simple coring a right semi-simple coring? We develope the basic essentials for a theory of semi-simple corings, giving a positive answer for the last question, as well as some information about the structure of semi-simple corings.