We begin this chapter presenting semisimple artinian rings. They are the rings for which all modules are projective, equivalently all modules are injective, or all modules are semisimple. These rings are finite direct products of rings of \(n\times n\) matrices over division rings (Artin-Wedderburn Theorem). This gives us the possibility of applying all the notions studied up to here in this book to the ring of matrices over a division ring. Then we consider the nilradical of a ring and right artinian rings, proving the Hopkins-Levitzki Theorem (a ring R with nilradical N is a right artinian ring if and only if R is right noetherian, N is nilpotent, and \(R/N\) is semisimple artinian).We study the radical of a module and the Jacobson radical of a ring. Finally we apply our results to group representations showing the relation between group representations and modules, and proving Maschke’s theorem.