A quick introduction to trace ideals

The well-known trace map on matrices can be generalized to a map on any module, M, over a commutative ring, R. Its image, the trace ideal of an R-module M, is the ideal generated by the homomorphic images of M in R. In this talk we will survey many recent developments in the theory of trace ideals within commutative algebra with far-reaching applications to the structure of endomorphism rings, Ext-rigidity in modules, and ideal-theoretic classifications of Arf rings.