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We investigate the notion of partial trace ideals, recently introduced by Maitra. We first establish several properties of partial trace ideals and give affirmative answers to questions posed by Maitra. We then study the invariant defined…
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…
The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We also give an…
Lower bounds on Hilbert-Samuel multiplicity are known for several types of commutative noetherian local rings, and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity. The first part of this…
The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 =…
We investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three…
A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding…
Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and…
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two…
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…
For a finite dimensional monomial algebra $\Lambda$ over a field $K$ we show that the Hochschild cohomology ring of $\Lambda$ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated $K$-algebra of…
We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1].…
We prove that level binomial edge ideals with regularity 2 and pseudo-Gorenstein binomial edge ideals with regularity 3 are cones, and we describe them completely. Also, we characterize level and pseudo-Gorenstein binomial edge ideals of…
In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of Macaulay's inverse system generalizing a recent result by J. Elias and M.E. Rossi. Then…
Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a…
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one…
We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the…
Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the…
Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in…