We characterize the well-covered property for theta-ring and ring graphs. Furthermore, we prove that Cohen–Macaulayness, pure shellability and pure vertex decomposability are equivalent for theta-ring graphs. Also, we give a combinatorial characterization of these graphs.

We study join-meet ideals associated with modular nondistributive lattices. We give a lower bound for the regularity and show that they are not linearly related.

Let $R$ be an integral domain. We investigate when $R$ has only finitely many star operations of finite type. We start by investigating some classes of integral domains with at most two star operations of finite type. Our aim is to generalize or find analogues of several characterizations of well-known classes of integral domains with at most two star operations. Next we concentrate on Mori domains with only finitely many star operations of finite type in a similar study done for Noetherian domains. We end by a section on pullbacks to enrich the literature with new and original examples and also to illustrate several results in the paper.

Let $\left(R,\mathfrak{𝔪},\mathbb{𝕜}\right)$ be a three-dimensional Cohen–Macaulay analytically unramified local ring and $I$ an $\mathfrak{𝔪}$-primary $R$-ideal. Write $X=\text{Proj}\left({\oplus }_{n\in \mathbb{N}}\overline{I^n}{t}^n\right)$. We prove some consequences of the vanishing of ${\text{H}}^2\left(X,{\mathcal{𝒪}}_X\right)$, whose length equals the constant term ${\overline{e}}_3(I)$ of the normal Hilbert polynomial of $I$. Firstly, $X$ is Cohen–Macaulay. Secondly, if the extended Rees ring $A:={\oplus }_{n\in \mathbb{Z}}\overline{I^n}{t}^n$ is not Cohen–Macaulay, and either $R$ is equicharacteristic or $\overline{I}=\mathfrak{m}$, then ${\overline{e}}_2(I)-{\text{length}}_R\left(\overline{I^2}/\left(I\overline{I}\right)\right)\ge 3$; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ${\mathbb{P}}_{\mathbb{𝕜}}^2$. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, ${\text{H}}_E^2\left(X,I^m{\mathcal{𝒪}}_X\right)=0$ for all integers $m$, where $E$ is the exceptional divisor in $X$. Finally, if additionally $R$ is regular and $X$ is pseudorational, then the adjoint ideals $\widetilde{I^n}$, $n\ge 1$ satisfy $\widetilde{I^n}=I\widetilde{I^{n-1}}$ for every $n\ge 3$. The last two results are related to conjectures of J. Lipman (1994).

Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group ${\text{GL}}_n\left(ℂ\right)$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules ${\text{Ext}}_S^{•}\left(S∕I,S\right)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusion of invariant ideals $I\supseteq J$, we compute the (co)kernel of the induced map ${\text{Ext}}_S^j\left(S∕I,S\right)\to {\text{Ext}}_S^j\left(S∕J,S\right)$ for all $j\ge 0$. As a consequence, we show that if an invariant ideal $I$ is unmixed, then the induced maps ${\text{Ext}}_S^j\left(S∕I,S\right)\to H_I^j\left(S\right)$ are injective, answering a question of Eisenbud, Mustaţă and Stillman in the case of Pfaffian thickenings. Finally, using our Ext computations and local duality, we verify an instance of Kodaira vanishing in the sense described in the recent work of Bhatt, Blickle, Lyubeznik, Singh and Zhang.

Let $\mathbb{𝕂}$ be any field. Given $n$ linear forms in $R=\mathbb{𝕂}\left[x_1,\dots ,x_k\right]$, some possibly proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $\left(n-2\right)$-fold products of these linear forms has linear graded free resolution. When no two of the linear forms considered are proportional, this result helps determining a complete set of generators of the symmetric ideal of $I$. Via Sylvester forms we can analyze from a different perspective the generators of the presentation ideal of the Orlik–Terao algebra of the second order; this is the algebra generated by the reciprocals of the products of any two (distinct) of the linear forms considered.

We give a criterion for rings with ${\mathfrak{𝔪}}^3=0$ which are obtained as connected sums of two other rings to have nontrivial totally acyclic modules.

Applications of Krull’s lemma and related transfinite principles can sometimes be eliminated by computational rules which govern deductions in corresponding formal theories. One thereby obtains elementary constructive arguments. The purpose of this note is to communicate an instance in the context of connected reduced rings.

For a partition $\lambda$ of $n$, let $I_{\lambda }^{\text{Sp}}$ be the ideal of $R=K\left[x_1,\dots ,x_n\right]$ generated by all Specht polynomials of shape $\lambda$. We show that if $R∕I_{\lambda }^{\text{Sp}}$ is Cohen–Macaulay then $\lambda$ is of the form either $\left(a,1,\dots ,1\right)$, $\left(a,b\right)$, or $\left(a,a,1\right)$. We also prove that the converse is true in the $\text{char}\left(K\right)=0$ case. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R∕I_{\left(n-3,3\right)}^{\text{Sp}}$ is not Cohen–Macaulay if and only if $\text{char}\left(K\right)=2$.