We perform a general study of the structure of locally compact modules over compactly generated abelian groups. We obtain a dévissage result for such modules of the form ‘compact-by-sheer-by-discrete’, and then study more specifically the sheer part. The main typical example of a sheer module is a polycontractible module, that is, a finite direct product of modules, each of which is contracted by some group element. We show that every sheer module has a ‘large’ polycontractible submodule, in some suitable sense. We apply this to the study of compactly generated metabelian groups. For instance, we prove that they always have a maximal compact normal subgroup, and we extend the Bieri–Strebel characterization of compactly presentable metabelian groups from the discrete case to this more general setting.

Restriction is a natural quasi-order on d-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field; namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property X. For instance, X admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g., homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property.

We prove that the infinite half-spin representations are topologically Noetherian with respect to the infinite spin group. As a consequence, we obtain that half-spin varieties, which we introduce, are defined by the pullback of equations at a finite level. The main example for such varieties is the infinite isotropic Grassmannian in its spinor embedding, for which we explicitly determine its defining equations.

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. Among the applications are precise connections between Ringel duality, split quasi-hereditary covers and double centralizer properties, constructions of split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of q-Schur algebras and a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring. They are shown to behave nicely under change of rings techniques.

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

In a 1968 issue of the Proceedings, P. M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a counterexample was however provided by A. Grams, who showed that every commutative domain with the ACCP is atomic, but not vice versa. This has led to the problem of finding a sensible (ideal-theoretic) characterisation of atomicity.

The question (explicitly stated on p. 3 of A. Geroldinger and F. Halter–Koch’s 2006 monograph on factorisation) is still open. We settle it here by using the language of monoids and preorders.

We construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$, i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine $r-1$-dimensional ${\mathbb {R}}$-variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$ . We study the set of intermediate rings between $R$ and $S$ . We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$. We prove that every $R$-homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? In this paper we provide, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens–Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

Let $R$ be a commutative ring. The regular digraph of ideals of $R$, denoted by $\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of $R$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$ whenever $I$ contains a nonzero divisor on $J$. In this paper, we study the connectedness of $\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in $\Gamma (R)$, whenever $R$ is a finite direct product of fields. Among other things, we prove that $R$ has a finite number of ideals if and only if $\mathrm {N}_{\Gamma (R)}(I)$ is finite, for all vertices $I$ in $\Gamma (R)$, where $\mathrm {N}_{\Gamma (R)}(I)$ is the set of all adjacent vertices to $I$ in $\Gamma (R)$.

Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

This work is concerned with the question of when a von Neumann regular ring is expressible as a directed union of Artinian subrings.

A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, implies that either or . In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

Let R be a commutative, Noetherian ring and let Q be the total quotient ring of R. We shall call B an intermediate ring if R ⊂ B ⊂ Q. In [S] it is proved, for an integral domain R, that if R ⊂ B ⊂ Rf where B is flat over R, then B is a finitely generated R-algebra. We observe that the result holds for any commutative, Noetherian ring where f is a non-zero divisor. Our proof [Theorem 1.1] is a little different and straight; it is given for completeness. The idea of the proof in [S] lies in finding an ideal I of R such that IB = B, and for any λ∈I, b∈B there exists m ≥ 1 such that λmb ∈ R. We shall show that even if an intermediate ring B is finitely generated R-algebra, there may not exist any ideal I of R such that IB = B, moreover, if B is not finitely generated R-algebra, we may have IB = B for some ideal I in R.