Generalized Gorenstein Arf rings (Q1741622)

Let $(R,\mathfrak{m})$ be a local one-dimensional Cohen-Macaulay ring with $Q(R)$ its total quotient ring. By view of \textit{C. Arf} [Proc. Lond. Math. Soc. (2) 50, 256--287 (1948; Zbl 0031.07002)] and \textit{J. Lipman} [Am. J. Math. 93, 649--685 (1971; Zbl 0228.13008)] it is called an Arf ring provided every integrally closed ideal is stable. An ideal $I \subset R$ is stable if $I^2 = xI$ for some $x \in I$ (see J. Lipman [loc. cit.] for the details). For the definition of a generalized Gorenstein ring see the paper by \textit{S. Gôto} et al., [``Characterization of generalized Gorenstein rings'', Preprint, \url{arXiv:1704.08901}]. The main subject of the paper is the study of generalized Gorenstein Arf rings. The main result is the following: \par Let $R$ denote a one-dimensional generalized Gorenstein local ring with a canonical ideal $I$ that contains a parameter ideal $(a)$ as a reduction. Then $R$ is an Arf ring if and only if $R$ has minimal multiplicity and the multiplicity of $S_{\mathcal{M}}$ is at most two for each maximal ideal of $S =R[x/a| x\in I]$. \par There are various applications. In particular, let $\Bbbk$ a field and let $R = \Bbbk[|t^{a_1}, \ldots, t^{a_l}|]$ be a generalized Gorenstein numerical semigroup ring. Then $R$ is an Arf ring if and only if $R$ is of minimal multiplicity and a certain numerical condition on the corresponding semigroup $H =\langle a_1,\ldots,a_l\rangle$ is satisfied. \par This generalizes a result by \textit{V. Barucci} and \textit{R. Fröberg} [J. Algebra 188, No. 2, 418--442 (1997; Zbl 0874.13018)] about Arf numerical semigroup rings.