Let $(k,\delta )$ be a differential field containing $\mathbf {Q}$. We prove that, in general, a radical differential ideal of the differential ring $k\{X,Y\}$ is not a differential ideal of finite type.

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e., invariant subrings $k[\mkern -2.75mu[ u,v]\mkern -2.75mu]^G$ of rings of formal power series under the action of a cyclic group. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

We introduce the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree though it is generated in a single degree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property.

In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the Boij-Soderberg decompositions of both the ideal and its quotient. We also give explicit formulae for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. Further still, we describe the associated primes, and we show that they are persistent.

We give bounds for Hilbert-Kunz multiplicity of products of two ideals and characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.

We give a short proof of a result due to Katz and West: Let $R$ be a Noetherian ring and $I_1,\ldots ,I_t$ ideals of $R$. Let $M$ and $N$ be finitely generated $R$-modules and $N' \subseteq N$ a submodule. For every fixed $i \ge 0$, the sets $$ \mathrm {Ass}_R(\mathrm {Ext}_R^i(M,N/I_1^{n_1}\cdots I_t^{n_t} N') ) $$ and $$ \mathrm {Ass}_R(\mathrm {Tor}_i^R(M,N/I_1^{n_1}\cdots I_t^{n_t} N') ) $$ are independent of $(n_1,\ldots ,n_t)$ for all sufficiently large $n_1,\ldots ,n_t$.

We call a directed acyclic graph a CI-digraph if a certain affine semigroup ring defined by it is a complete intersection. We show that if $D$ is a 2-connected CI-digraph with cycle space of dimension at least 2, then it can be decomposed into two subdigraphs, one of which can be taken to have only one cycle, that are CI-digraphs and are glued together on a directed path. If the arcs of the digraph are the covering relations of a poset, this is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner. The decomposition result shows that CI-digraphs can be easily recognized.

We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set $\mathrm {Star}(R)$ of star operations on $R$ as the product of $\mathrm {Star}(T)$, as $T$ ranges over a family of overrings of $R$ with special properties. We then apply this method to study the set of star operations on a Prufer domain $R$, in particular the set of stable star operations and the star-class groups of $R$.