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Commutative ring theorists have mainly concentrated in the study of Cohen-Macaulay rings and modules in the last fifty years. Understanding Cohen-Macaulay rings are important since they lie within the intersection of algebraic geometry, algebraic topology, and commutative algebra. There are many Cohen-Macaulay rings which are not Gorenstein, but sufficiently good next to Gorenstein rings; for example, almost Gorenstein rings and generalized Gorenstein rings. These naturally cover the class of Gorenstein rings and fill the gap in between Cohen-Macaulay and Gorenstein properties.

In 1971 Lipman studied a class of local rings called Arf rings; these rings were initially introduced by Arf in 1949 pertaining to a certain classification of curve singularities. Although the definition of an Arf ring is technical, examples of such rings are abundant in the literature, for example, a one-dimensional local domain of multiplicity two is Arf.

In this talk we discuss several examples and motivations for studying Arf rings that are almost Gorenstein and generalized Gorenstein, and give a characterization of such rings.

This talk is based on the joint work with Olgur Celikbas, Shiro Goto, and Naoki Endo.