S. Kumashiro

On embeddings of rings into their canonical modules

Tran Nguyen An, Shinya Kumashiro, Mouma Samanta

We study embeddings of Cohen--Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summa…

Prescribed Initial Behavior of $μ (I^{k})$

Reza Abdolmaleki, Shinya Kumashiro

It is well known that for every graded ideal $I$ , the numbers of minimal generators $μ (I^{k})$ of its powers of $I$ are eventually increasing. However, its initial behavior can be su…

The Non-Pure Dual Exchange Property in Low Dimensions

Reza Abdolmaleki, Shinya Kumashiro

We investigate monomial ideals satisfying the non-pure dual exchange property, a notion introduced in connection with componentwise polymatroidal ideals. Our contributions are twof…

Quasi-factorially closed subalgebras of Laurent polynomial rings

Shinya Kumashiro, Takanori Nagamine

Let $R$ be a domain and $B = R [x_{1}^{\pm 1}, \dots, x_{n}^{\pm 1}]$ the Laurent polynomial ring over $R$ . In this paper we study pre-factorially closed (pfc) and quasi-factorially closed (…

On Partial Trace Ideals

Souvik Dey, Shinya Kumashiro

We investigate the notion of partial trace ideals, recently introduced by Maitra. We first establish several properties of partial trace ideals and give affirmative answers to ques…

Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings

Ela Celikbas, Olgur Celikbas, Naoki Endo +1

We study finitely generated modules of minimal multiplicity, a notion introduced by Puthenpurakal that extends the classical concept of minimal multiplicity from rings to modules.…

math.AC2025

Characterization of almost Gorenstein rings in terms of the trace ideal

Ryotaro Isobe, Shinya Kumashiro

We provide a characterization of one-dimensional almost Gorenstein rings in terms of the trace ideal. As an application, we investigate the almost Gorenstein property of certain $\…

math.AC2025

Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings

Shinya Kumashiro, Sora Miyashita

Recent work by Miyashita and Varbaro classified the canonical traces of Stanley--Reisner rings that are Gorenstein on the punctured spectrum, under the Cohen--Macaulay assumption.…

math.AC2025

Full-Trace Modules

Ela Celikbas, Olgur Celikbas, Jürgen Herzog +1

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules wh…

math.AC2024

Defining ideals of Cohen-Macaulay fiber cones

Reza Abdolmaleki, Shinya Kumashiro

Let $A$ be a commutative Noetherian local ring with maximal ideal $m$ , and let $I$ be an ideal. The fiber cone is then an image of the polynomial ring over the residue f…

math.AC2023★ 1 cited

Traces of semi-invariants

Ela Celikbas, Jürgen Herzog, Shinya Kumashiro

This article investigates the traces of certain modules over rings of invariants associated with finite groups. More precisely, we provide a formula for computing the traces of arb…

math.AC2023

Nearly Gorenstein local rings defined by maximal minors of a $2 \times n$ matrix

Shinya Kumashiro, Naoyuki Matsuoka, Taiga Nakashima

We investigate the nearly Gorenstein property of a local ring defined by the maximal minors of a specific $2 \times n$ matrix with entries in the formal power series ring $k[[X_1,…

math.AC2022★ 2 cited

On generalized Gorenstein local rings

Shiro Goto, Shinya Kumashiro

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus b…

On a generalized Auslander-Reiten conjecture

Souvik Dey, Shinya Kumashiro, Parangama Sarkar

It is well-known that the generalized Auslander-Reiten condition (GARC) and the symmetric Auslander condition (SAC) are equivalent, and (GARC) implies that the Auslander-Reiten con…

On the sectional genera and Cohen-Macaulay rings

Shinya Kumashiro, Hoang Le Truong, Hoang Ngoc Yen

We explore the behavior of the sectional genera of certain primary ideals in Noetherian local rings. In this paper, we provide characterizations of a Cohen-Macaulay local ring in t…

The set of trace ideals of curve singularities

Toshinori Kobayashi, Shinya Kumashiro

This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed…

Upper bound on the colength of the trace of the canonical module in dimension one

Jürgen Herzog, Shinya Kumashiro

We study the upper bound of the colength of trace of the canonical module in one-dimensional Cohen-Macaulay rings. We answer the two questions posed by Herzog-Hibi-Stamate and Koba…

The irreducible multiplicity and Ulrich modules

Tran Nguyen An, Shinya Kumashiro

In this paper, we give a relation between the Hilbert multiplicity and the irreducible multiplicity. As an application, we characterize Ulrich modules in term of the irreducible mu…

When are trace ideals finite?

Shinya Kumashiro

In this paper, we study Noetherian local rings $R$ having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral clos…

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

Jürgen Herzog, Shinya Kumashiro, Dumitru I. Stamate

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We inv…

Integrally closed ideals of reduction number three

Shinya Kumashiro

In a Cohen-Macaulay local ring $(A, m)$ , we study the Hilbert function of an integrally closed $m$ -primary ideal $I$ whose reduction number is three. With a m…