交换代数

For a commutative ring $R$ with identity, the zero-divisor graph of $R$, denoted $\Gamma(R)$, is the graph whose vertices are the non-zero zero divisors of $R$ with two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In…

Let $(R,\fm)$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z) = \nu(\Ext^{g-\inf C}_R(Z,C))\mu^{\depth C}_R(\mathfrak{m}, C)$ for a homologically finite and bounded complex $Z$ with finite…

To every simple toric ideal $I_T$ one can associate the strongly robust simplicial complex $\Delta _T$, which determines the strongly robust property for all ideals that have $I_T$ as their bouquet ideal. We show that for the simple toric…

We provide the necessary and sufficient conditions for the edge-binomials of the tree forming a $d$-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated…

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $M,X$ two $R$-modules. In this paper, we study the Artinianness and cofiniteness of the module $H^i_{\Phi}(M,X)$ which is an extension of the generalized local…

Let $R$ be a commutative ring with identity $1\neq 0$. In this paper, we continue the study started in [10] concerning when the extended zero-divisor graph of $R$, $\overline{\Gamma}(R)$, is complemented. We also study when…

Let $(R,\fm)$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or…

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Suppose there exists be a non-zero $A$ module $M$ of finite length and finite projective dimension such that $\ell\ell(M)$, the Lowey length of $M$, is equal to…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $N$ a finite $R$-module. We prove that \cite[Theorem 2.1]{Mel} and \cite[Proposition 3.3 (i)$\Leftrightarrow$(ii)]{B1} are true for any Serre…

The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as $\operatorname{Ker} L$ is a Mathieu-Zhao space where $L$ is…

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation…

In this paper, we study the class of affine semigroup generated by integral vectors, whose components are in generalised arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on…

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such…

Let $K$ be a field with ${\rm char}(K)=0$. For a partition $\lambda$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. These ideals have been…

F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity. One would like to have a variant that rather detects F-rationality, and there are two theories that…

Ananyan and Hochster proved the existence of a function $\Phi(m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi(m,d)$. Relatedly,…

We compute the finite generation ideal for Daigle and Freudenburg's counterexample to Hilbert's fourteenth problem. This ideal helps to understand how far the ring of invariants is from being finitely generated. Our calculations show that…

Connected bipartite graphs whose binomial edge ideals are Cohen--Macaulay have been classified by Bolognini et al. In this paper, we compute the depth, Castelnuovo--Mumford regularity, and dimension of the generalized binomial edge ideals…

This paper presents an $\mathcal{E}$-derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite $K$-$\mathcal{E}$-derivation of polynomial algebras in two variables over…

In this article, we prove necessary conditions for Hibi rings to satisfy Green-Lazarsfeld property $N_p$ for $p=2$ and $3$. We also show that if a Hibi ring satisfies property $N_4$, then it is a polynomial ring or it has a linear…