Commutative algebra 2 (Rings, ideals, modules)

This lecture is a review of rings, ideals, and modules, serving as a foundation for a course on commutative algebra. The speaker begins by recalling the definition of a ring, noting that there are four inequivalent definitions in the literature, and clarifies that in this course rings are commutative and have an identity. He provides examples of non-commutative rings, such as matrix rings, quaternions, group rings, and rings of differential operators, to illustrate the diversity of rings. He then discusses the issue of rings without identity, explaining why analysts often consider them, and shows how adding an identity corresponds to one-point compactification in topology. The lecture proceeds to define ideals as kernels of homomorphisms and gives examples like nZ and polynomial ideals, leading to quotient rings. Finally, modules are introduced as generalizations of vector spaces, with examples including abelian groups, vector spaces, and modules over polynomial rings, and the relationship between ideals and modules is explored. The lecture concludes by drawing analogies between groups and rings, such as normal subgroups corresponding to ideals.