Τhe resarch project FiCiHAlgIA is funded by the Hellenic Foundation for Research and Innovation (HFRI) under the "First Call for H.F.R.I. Research Projects to support Faculty members and Researchers". The objective of the project is to study certain problems in algebra and, in particular, the theory of modules over groups and rings, using homological methods and techniques. The proposed research will lead to a deeper understanding of the structure of groups and rings and improve our knowledge of the way these algebraic objects are used in geometry and representation theory.