Weakly Arf rings

Abstract

In this paper, we introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.

Topics on strict closure of rings

“…F -pure rings satisfying (S 2 )) are strictly closed. The reader may consult with [5,2,3] about further study of strictly closed rings.…”

“…As is mentioned in [5], O. Zariski conjectured that the strict closure of R in R coincides with the Arf closure of R. As for this conjecture, O. Zariski proved the if part, and J. Lipman gave an affirmative answer for the converse, when R contains a field ([5, Proposition 4,5, Theorem 4.6]). Until this was settled in [2,Theorem 4.4] with full generality, it seems that Zariski's conjecture has been open for more than one half of a century.…”

“…Recently, the authors of [2] introduced the notion of weakly Arf rings, towards a theory of higher dimensional Arf rings, by weakening the defining conditions of Arf rings. For an arbitrary commutative ring R, the ring R is called weakly Arf, if R satisfies condition (2) in the above definition.…”

“…Recently, the authors of [2] introduced the notion of weakly Arf rings, towards a theory of higher dimensional Arf rings, by weakening the defining conditions of Arf rings. For an arbitrary commutative ring R, the ring R is called weakly Arf, if R satisfies condition (2) in the above definition. As J. Lipman predicted in [5] and as the authors of [2] are now developing the theory of weakly Arf rings, there might have a strong connection between the strict closedness and the weak Arf property.…”

“…For an arbitrary commutative ring R, the ring R is called weakly Arf, if R satisfies condition (2) in the above definition. As J. Lipman predicted in [5] and as the authors of [2] are now developing the theory of weakly Arf rings, there might have a strong connection between the strict closedness and the weak Arf property. In fact, for an arbitrary commutative ring R, there exists the weakly Arf closure R a of R which is a smallest weakly Arf ring between R and R, and the following holds.…”