Modules over a ring are algebraic structures that generalize vector spaces by allowing scalars from a ring instead of a field. They consist of an abelian group equipped with an action by the ring, which means that you can add and scale elements using the ring's operations. This setup connects closely to projective modules, which are direct summands of free modules and help in understanding how modules can behave in terms of decomposition and extension properties.
congrats on reading the definition of Modules over a Ring. now let's actually learn it.