Translates of Polynomials

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We classify linear systems of parameters of $S/I$ over any field $K$ using linear algebra and show explicit linear systems of parameters when $K$ has at least $n$ elements. If $I(G)$ is the edge ideal of a perfect graph $G$, a cycle or the complement of a cycle, we show that $S/I(G)$ has a 0-1 linear system of parameters, and for graphs with independence number equal to $2$, we characterize when $S/I(G)$ has a 0-1 linear system of parameters.

math.AC↗

arXiv2026

Primality of dilated closed path polyominoes

In this paper we define a novel product on polyominoes called the tensor product of polyominoes. Using a special case of this product which we name a dilation, we construct a new class of polyominoes called the dilated closed paths. This class is a ''thick'' generalization of Cisto and Navarra's class of closed paths (arXiv:2006.13935). We prove the Zig-Zag Walk Conjecture for this new class thereby fully characterizing primality for its associated polyomino ideals.

math.AC↗

arXiv2026

Reduced and coreduced modules with respect to inverse families of ideals

Let R be a commutative ring and Phi an inverse family of ideals with greatest proper member L. We introduce and study Phi reduced and Phi coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized torsion and completion functors associated with Phi are determined by L. We establish characterizations and closure properties of these modules and obtain a Greenlees May type adjunction and a Matlis Greenlees May type characterization. When Phi is a system of ideals, we further investigate the radicality of the generalized torsion functor and derive associated torsion theories on suitable Serre subcategories. These results provide a module theoretic framework for studying generalized torsion and completion with respect to families of ideals

math.AC↗