We will aim to have a geometric picture of such a ring R, a picture that is modelled on the way in which an affine algebraic variety is associated to a commutative ring. The simple R-modules play the role of the points. These points are in bijection with the maximal two-sided ideals of R. They are also the closed points of the prime spectrum Spec(R) endowed with the analog of the Zariski topology. The inclusion Z R of the center in R induces a map f: Spec(R)  Spec(Z) and this is a homeomorphism on a non-empty open set.

A more detailed analysis of this map requires us to develop some structure theory for R, and these results are modelled on the commutative results. We will encounter the notion of a maximal order (the analog of an integrally closed ring, or normal variety), the idea of the ring of fractions of a prime noetherian ring, Azumaya algebras, and finite dimensional algebras (because these control the fibers of the map f). We will say a little about representations of quivers.

Important motivating examples include skew group rings R=A*G constructed from the action of a finite group on the coordinate ring A of an affine algebraic variety X. In this case the center of R is the coordinate ring of the quotient/orbit variety X/G and the module theory of R is closely related to the geometry of the (usually singular) variety X/G and its (possible) resolutions. We will look closely at the Kleinian singularities, these being the varieties X/G when X is C², the 2-dimensional complex vector space, and G is a finite subgroup of SL(2,C). If we have time we will also consider the deformations of these.

In summary, the course will be a mixture of noncommutative algebra, commutative algebra, algebraic geometry, and representation theory. To some extent the choice of topics is flexible and will depend on the audience.