The present paper provides a broad study of Arf rings, along with introducing and studying the concept of weakly Arf rings. The introduction of the paper begins by sketching the origin of Arf rings with roots in the geometry of curves.
For example, the authors give an affirmative answer to a conjecture of Oscar Zariski concerning the equality of the Arf closure and the strict closure. Also, the (weakly) Arf property of certain class of rings, including Nagata idealization, invariant rings, semigroup rings, fibre products, determinantal rings, cyclic pure descends, polynomial extensions and certain localization of certain blow-ups are stdudied.
An Arf ring is Noetherian semilocal ring \(A\) such that \(A_M\) is a Cohen-Macaulay \(1\)-dimensional local ring for all maximal ideal \(M\), and such that

1.

Each integrally closed ideal of positive graded has a principal reduction.

2.

If for \(x,y,z\in A\) with \(x\) a non-zero divisor, \(y/x,z/x\) are integral over \(A\) then \(yz/x\in A\).

The authors of the present paper define for a Noetherian ring \(A\) to be a weakly Arf ring, provided the condition \(2\) above is only required for \(A\) to be satisfied (here \(A\) is not necessarily semilocal and with no Cohen-Macaulay or dimension \(1\) assumption). Thus integrally closed domains and depth zero domains are examples of weakly Arf rings in this setting. For a one dimensional Cohen-Macaulay local ring \(A\) such that its residue field is infinite, or it is analytically irreducible, being weakly Arf and Arf are equivalent. However, the authors discuss examples of one dimensional Cohen-Macaulay local weakly Arf rings which are not Arf.