1. Recognize the maximal ideals and the prime ideals of a ring.
2. Identify the radical of a ring.
3. Know the properties of noetheriens rings.
4. Be familiar with all types of modules (free, finite type, torsion, torsion and noetherien)
5. Find the rational canonical form and the Jordan form of a matrix.
Rings : Revision and examples on the rings, maximal ideal and Jacobson radical, prime ideal first and nilradical, product of ideals and power of an ideal, ideal generated by a subset and ideal of finite type.
Modules : Definition and examples of modules, submodules and quotient module, homomorphism of modules, isomorphism theorems of modules, products of modules, direct sums of submodules, annihilator of a module, free module, module of finite type, torsion and torsion free module.
Noetherian module and noetherian ring : Definition, example, and properties.
Artinien module and artinien ring : Definition, example and properties.
Reduction of a matrix : modules of finite type and of torsion on a principal ideal ring, invariant subspaces, companion matrix and rational canonical form of a matrix, Jordan matrix and Jordan form of a matrix, diagonalization and trigonalization of matrices.