The divisor sequence of an irreducible element (atom) $a$ of a reduced monoid $H$ is the sequence ${(s_n)}_{n\in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. We investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull–Remak–Schmidt property fails.

We study the quantitative behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $𝒪(d,2)$ conform to Ein, Erman, and Lazarsfeld’s normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.

We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of $R$ in countable multiplicative subsets. We also show that if the spectrum of $R$ is of cardinality less than $\kappa$, where $\kappa$ is an uncountable regular cardinal, then every flat $R$-module is a transfinite extension of flat modules with less than $\kappa$ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat $R$-module is quite flat. We show that all von Neumann regular rings and all $S$-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.

For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of B. Ulrich, in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.

Given a 0-dimensional affine $K$-algebra $R=K[x_1,\dots ,x_n]∕I$, where $I$ is an ideal in a polynomial ring $K[x_1,\dots ,x_n]$ over a field $K$, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether $R$ is a complete intersection at a maximal ideal, whether $R$ is locally a complete intersection, and whether $R$ is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of $I$ form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field $K$. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

If $R\subseteq S$ is a ring extension of commutative unital rings, the poset $[R,S]$ of $R$-subalgebras of $S$ is called catenarian if it verifies the Jordan–Hölder property. This property has already been studied by Dobbs and Shapiro for finite extensions of fields. We investigate this property for arbitrary ring extensions, showing that many types of extensions are catenarian. We reduce the characterization of catenarian extensions to the case of field extensions, an unsolved question at this time.

We propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection $R$. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose zeroth homologies are the residue field of $R$. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The statement about the first Koszul homology is shown to be equivalent to the monomial conjecture and thus follows from its validity.

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I\subseteq R$ defining compressed rings with socle $k(-s)\oplus k(-2s+1)$, where $s\ge 3$ is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman (J. Commut. Algebra 11:3 (2019), 325–339). We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we give bounds on the minimal number of generators $\mu (I)$ of $I$ depending only on $s$; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all $s\ge 3$. Finally, we study the Tor-algebra structure of $R∕I$. It is shown that these rings have Tor algebra class $G(r)$ for $s\le r\le 2s-1$. Furthermore, we produce ideals $I$ for all $s\ge 3$ and all $r$ with $s\le r\le 2s-1$ such that $\text{Soc}(R∕I)=k(-s)\oplus k(-2s+1)$ and $R∕I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov (J. Pure Appl. Algebra 216:11 (2012), 2489–2506).

Let $R$ be a commutative ring with identity and let $\mathcal{𝒬}$ be the set of finitely generated semiregular ideals of $R$. A $\mathcal{𝒬}$-torsion-free $R$-module $M$ is called a Lucas module if ${\text{Ext}}_R^1(R∕J,M)=0$ for any $J\in \mathcal{𝒬}$. Moreover, $R$ is called a DQ ring if every ideal of $R$ is a Lucas module. We prove that if the small finitistic dimension of $R$ is zero, then $R$ is a DQ ring. In terms of a trivial extension, we construct a total ring of quotients of the type $R=D\propto H$ which is not a DQ ring. Thus in this case, the small finitistic dimension of $R$ is not zero. This provides a negative answer to an open problem posed by Cahen et al.