磯部 遼太郎

We provide a certain direct-sum decomposition of reflexive modules over (one-dimensional) Arf local rings. We also see the equivalence of three notions, say, integrally closed ideals, trace ideals, and reflexive modules of rank one (i.e., divisorial ideals) up to isomorphisms in Arf rings. As an application, we obtain the finiteness of indecomposable reflexive modules, up to isomorphism, for analytically irreducible Arf local rings.

In 1971, Lipman (Am J Math 93:649-685, 1971) introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns construction of the closure, and the second one is the study regarding the question of whether the strict closedness is inherited under flat homomorphisms. We finally handle the question of when the Arf closure coincides with the strict closure. Examples are explored to illustrate our theorems.

<title>Abstract</title>The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline1.png" /><tex-math> $R \times _T S$ </tex-math></alternatives></inline-formula> of Cohen–Macaulay local rings <italic>R</italic>, <italic>S</italic> of the same dimension <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline2.png" /><tex-math> $d&gt;0$ </tex-math></alternatives></inline-formula> over a regular local ring <italic>T</italic> with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000843952000051X_inline3.png" /><tex-math> $\dim T=d-1$ </tex-math></alternatives></inline-formula> is an almost Gorenstein ring if and only if so are <italic>R</italic> and <italic>S</italic>. In addition, the other generalizations of Gorenstein properties are also explored.

This paper studies Ulrich ideals in hypersurface rings. A characterization of Ulrich ideals is given. Using this characterization, we construct a minimal free resolution of an Ulrich ideal concretely. We also explore Ulrich ideals in a hypersurface ring of the form R= k[[ X, Y]]/(f). (C) 2020 Elsevier B.V. All rights reserved.

The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are 2-AGL rings. We also explore Ulrich ideals in 2-AGL rings, mainly two-generated ones. (C) 2020 Elsevier Inc. All rights reserved.

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties. (C) 2019 Elsevier B.V. All rights reserved.

This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and GGL rings are analyzed.