arXiv subjects

Sholastica Luambano

Publications and source records attributed to Sholastica Luambano.

3 recordsLinked to original sources

arXiv2026

Reduced and coreduced modules with respect to inverse families of ideals

Let $R$ be a commutative ring and $Φ$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $Φ$-reduced and $Φ$-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 $Φ$ are determined by $\mathcal{L}$. We establish characterizations and closure properties of these modules and obtain a Greenlees-May type adjunction and a Matlis-Greenlees-May type characterization. When $Φ$ 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↗

arXiv2026

Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$.

math.AC↗

arXiv2026

Locally prime modules

For a commutative unital ring $R$ with fixed ideals $I$ and $J$, we introduce and study $I$-prime $R$-modules and $(I, J)$-prime $R$-modules together with their duals $I$-coprime $R$-modules and $(I,J)$-coprime $R$-modules respectively. We employ category-theoretic techniques to reveal their structural properties. Our main results are versions of the Greenlees-May Duality and the Matlis-Greenlees-May Equivalence to the setting of these prime and coprime modules. This generalizes work on $I$-reduced modules and $I$-coreduced modules. We demonstrate that these ``locally prime" modules serve as a tool for studying the classical ``globally prime" modules, creating a bridge between local and global primality.

math.AC↗