Introduction to Homological Algebra for Quantaloids: Semisimple Modules over a Quantaloid

This paper is a step towards a homological classification of quantales and quantaloids by characterizing semisimple quantales and quantaloids via their modules. We prove that semisimple quantales and quantaloids are quite similar to semisimple rings via their homological dimensions. More precisely, we study simple and semisimple modules and we connect them to the notions of artianity and noetherianity. We proved that simple modules are cyclic, establishing a connection with maximal congruences. Furthermore, we demonstrated that the structure of semisimple quantaloid modules closely resembles that of semisimple modules over a ring. Specifically, the ascending and descending chain conditions coincide, along with the occurrence of split short exact sequences and other related properties.