Let $G$ be an additive abelian group. A sequence $S=g_1\cdot \dots \cdot g_{\ell }$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are ${𝜀}_1,\dots ,{𝜀}_{\ell }\in \{1,-1\}$ such that ${𝜀}_1{g}_1+\dots +{𝜀}_{\ell }{g}_{\ell }=0$. We first characterize (in terms of $G$) when the monoid ${\mathcal{ℬ}}_{\pm }(G)$ of plus-minus weighted zero-sum sequences is Mori, respectively, Krull, respectively, finitely generated. After that, we study the isomorphism problem and the characterization problem for monoids of plus-minus weighted zero-sum sequences.

We show that codimension three Artinian Gorenstein sequences are log-concave and that there are codimension four Artinian Gorenstein sequences that are not log-concave. We also show the log-concavity of level sequences in codimension two.

Let $R=k[x_1,\dots ,x_n,\dots ]$ be the infinite variable polynomial ring equipped with the natural ${\mathfrak{𝔖}}_{\infty }$ action, where $k$ is a field of characteristic zero. In recent work, Nagpal and Snowden [5] gave an indirect proof that the ${\mathfrak{𝔖}}_{\infty }$-ideal generated by ${(x_1-x_2)}^{2n}$ is not ${\mathfrak{𝔖}}_{\infty }$-prime. In this paper, we give a direct proof, with explicit elements. We further formulate some conjectures on possible generalizations of the result.

We characterize the Gorenstein property for endomorphism rings of a fractional ideal on a curve singularity by using properties of the ideal. Moreover, the Gorenstein algebroid curves with only Gorenstein integral extensions are classified.

Let $(R,\mathfrak{𝔪})$ be a Noetherian local ring and $M$ be a finitely generated $R$-module. In this paper, we give precise formulas computing all the partial Euler–Poincaré characteristics of Koszul complexes of the idealization $R⋉M$ with respect to an almost p-standard system of parameters (a strict subclass of d-sequences) in terms of multiplicities of certain subquotients of $R$ and $M$. As an application, we clarify the invariants $p_k(R⋉M)$ of the idealization $R⋉M$ that were introduced by N. T. Cuong and Khoi (1996).

We establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen–Macaulay modules. Let $R$ be a commutative Noetherian ring (not necessarily local) with identity, and let $\mathfrak{𝔞}$ be a proper ideal of $R$. The notions of $\mathfrak{𝔞}$-relative dualizing modules and $\mathfrak{𝔞}$-relative big Cohen–Macaulay modules are introduced. With the help of $\mathfrak{𝔞}$-relative dualizing modules, we establish the global analogue of duality on the subcategory of Cohen–Macaulay modules in local algebra. Lastly, we investigate the behavior of the subcategory of $\mathfrak{𝔞}$-relative Cohen–Macaulay modules and $\mathfrak{𝔞}$-relative generalized Cohen–Macaulay modules under Foxby equivalence.

The goal of this paper is to show that if $R$ is an unramified hypersurface, if $M$ and $N$ are finitely generated $R$-modules, and if ${\text{Ext}}_R^n(M,N)=0$ for some $n\le {\text{grade}}_{R}M$, then ${\text{Ext}}_R^i(M,N)=0$ for $i\le n$. A corollary to this says that if $M\ne 0$, then ${\text{Ext}}_R^i(M,M)\ne 0$ for $0\le i\le {\text{grade}}_{R}M$. We also give an extension of this result to complete intersections. These results are related to a question of Jorgensen and results of Dao.

We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set for the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen–Macaulayness and Castelnuovo–Mumford regularity.