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We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders,…

We present a formalization of Gr\"obner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gr\"obner basis theory, including…

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is…

This paper considers generalizations of certain arithmetic complexes appearing in the work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued…

Let $A$ be a commutative ring with unity, and $M$ a finitely generated $A$-module. In 1971, Morris Orzech showed that any surjective $A$-module homomorphism from a submodule of $M$ to $M$ must be an isomorphism. We give a constructive proof…

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual…

Let $I$ be the ideal generated by the maximal minors of a matrix of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit…

We develop a semigroup-theoretic analogue of liaison for relative ideals of a numerical semigroup. Two parallel linkage notions are proposed: a theory based on translates of the semigroup and a theory based on translates of the canonical…

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among…

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k)$, the natural surjective map between the groups of units…

In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact…

This article discusses ramification and the structure of relative K\"ahler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the…

The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring…

We show that any polarization of an Artin monomial ideal defines a triangulated ball. This proves a conjecture of A.Almousa, H.Lohne and the first author. Geometrically, polarizations of ideals containing $(x_1^{a_1}, \ldots, x_n^{a_n})$…

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus…

In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of…

We provide exact formulas for the depth of the quotient ring of powers of the edge ideal of an increasing weighted path.

We study reducing invariants of modules related to certain homological properties. For modules of finite reducing projective dimension, we establish grade inequalities. We prove that if $\mathbb{P}$ is the (uniform) Auslander condition, or…

We investigate the finiteness of the set of associated primes for local cohomology modules $H_I^{i}(J)$ of an ideal $J$ generated by an $R$-sequence, through the comparison of $H_I^{d+1}(J)$ and $H_I^d(R/J)$, where $d =…

We compare two classes of polynomial automorphisms, strongly nilpotent and Pascal finite. We conclude that every strongly nilpotent automorphism is a Pascal finite one, but not vice versa. We observe that Nagata's automorphism is Pascal…