Let $\mathbb{𝔸}$ be a Dedekind domain and $T$ an endomorphism of a finitely generated projective $\mathbb{𝔸}$-module. If $T$ is an $s$-th power in ${\text{End}}_{\mathbb{𝔸}}(M)$ for $s$ ranging over an infinite set $\mathcal{𝒮}$ of positive integers, then (a) $T$ decomposes as a direct sum of the zero operator and an invertible operator on a summand of $M$ and (b) that summand is semisimple or of finite order if $\mathcal{𝒮}$ is appropriately large (what this means depends on the structure of the additive and multiplicative groups of $\mathbb{𝔸}$). This generalizes a result of Cavachi’s to the effect that the only nonsingular integer matrix that is an $s$-th power in $M_n(\mathbb{Z})$ for all $s$ is the identity.

Explicit minimal generators for Fulton’s Schubert determinantal ideals are determined along with some implications.

Shitov recently gave counterexamples over the real and complex field to Comon’s conjecture that the symmetric tensor rank and tensor rank of a symmetric tensor are the same. In this paper, we show that an analog of Comon’s conjecture for the $G$-stable rank introduced by Derksen is true: the symmetric $G$-stable rank and $G$-stable rank of a symmetric tensor are the same over perfect fields. We also show that the log-canonical threshold of a complex singularity is bounded by the $G$-stable rank of the defining ideal.

We give an algebraic characterization of the affine $3$-space over an algebraically closed field of arbitrary characteristic. We use this characterization to reformulate the following question. Let

$$A=k[X,Y,Z,T]∕(XY+Z^{p^e}+T+T^{sp}),$$

where $p^e\nmid sp$, $sp\nmid p^e$, $e,s\ge 1$ and $k$ is an algebraically closed field of positive characteristic $p$. Is $A=k^{[3]}$? We prove some results on $\text{ML}$ and ${\text{ML}}^{\ast }$ invariants and use them to prove a special case of the strong cancellation of $k^{[2]}$.

Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring extension ${\text{Int}}^{\text{R}}(D)$ of $D$. For a valuation domain $V$, we characterize when ${\text{Int}}^{\text{R}}(V)$ is a Prüfer domain and when ${\text{Int}}^{\text{R}}(V)$ is a Bézout domain. We also extend the classification of when ${\text{Int}}^{\text{R}}(D)$ is a Prüfer domain.

Let $M$ be a cancellative and commutative monoid (written additively). The monoid $M$ is atomic if every noninvertible element can be written as a sum of irreducible elements (often called atoms in the literature). Weaker versions of atomicity have been recently introduced and investigated, including the properties of being nearly atomic, almost atomic, quasiatomic, and Furstenberg. In this paper, we investigate the atomic structure of lattice monoids, i.e., submonoids of a finite-rank free abelian group, putting special emphasis on the four mentioned atomic properties.

Let $(R,\mathfrak{𝔪})$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z)=\nu ({\text{Ext}}_R^{g-\text{inf}C}(Z,C)){\mu }_R^{\text{depth}C}(\mathfrak{𝔪},C)$ for a homologically finite and bounded complex $Z$ with finite ${\text{G}}_C$-dimension $g$. Additionally, we prove that if ${\text{Ext}}^i(M,N)=0$ for sufficiently large $i$, while ${\text{id}}_R{\text{Ext}}^i(M,N)$ remains finite for all $i$, then both ${\text{pd}}_{R}M$ and ${\text{id}}_{R}N$ are finite when $M$ and $N$ are finitely generated $R$-modules. These findings extend the recent results of Ghosh and Puthenpurakal (Algebr. Represent. Theory 27:1 (2024), 639–653), addressing the questions as presented in their Questions 3.9 and 4.2.

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a noncoherent GCD domain, answering a question of Patankar in the negative concerning characterizations of $F$-coherent rings. This recovers and extends a result of Glaz using tight closure methods.

Using the recent results on square-free Gröbner degenerations by Conca and Varbaro, we prove that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal ${\mathrm{in}}_{<}\left(I\right)$ is square-free and ${\beta }_0\left(I\right)={\beta }_0\left({\mathrm{in}}_{<}\right(I\left)\right)$, then $I$ is a componentwise linear ideal if and only if ${\mathrm{in}}_{<}\left(I\right)$ is a componentwise linear ideal. In particular, if furthermore one of $I$ and ${\mathrm{in}}_{<}\left(I\right)$ is componentwise linear, then their graded Betti numbers coincide.