Noncommutative Rings, Modules, Categories, and Commutative Monoids

The content of the course will be:

• Rings, division rings

• Categories, functors

• Modules, bimodules, submodules and quotients

• Natural transformation of functors

• Sets of generators, maximal subodules

• Commutative monoids, pre-ordered groups, positive cones

• The monoid $V(\mathcal{C})$, discrete valuations, Krull monoids

• Semisimple rings and modules, free rings and free algebras, ranks of free modules

• Projective modules and radical, projective covers, injective envelopes

• The monoid $V(R)$, the Grothendieck group $K_0(R)$

• Direct limits of projectives, inverse limits of injectives

No particular previous knowledge apart from the elementary notions of Algebra that every PhD student in Mathematics knows. No particular prerequisites are necessary.