We show that every polynomial overring of the ring $\mathrm{Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\mathbb{\widehat Z}$, the profinite completion of $\mathbb Z$ with respect to the fundamental system of neighbourhoods of $0$ consisting of all non-zero ideals of $\mathbb Z$.

In this paper we get two new classes of regular sequences in the polynomial ring over the field of complex numbers.

This article establishes a concrete relation between order ideals of minimal generators and annihilator ideals. For a regular local ring $R$ and ideal $I$ the authors construct an $R$-module $M$ with minimal generator having $I$ as order ideal. Further, it is shown that most variability in ideal–theoretic behavior of such order ideals is exhibited by modules of projective dimension~one. The authors ``introduce'' the concept of $*$-orthogonality and use their syzygy theorem to show constraints on the size and height of a $*$-orthogonal set in a given finitely generated non-free module. The paper contains an application of the theory of order ideals to the binomial behavior of syzygy rank.

In this paper, we will introduce a new norm map on almost Dedekind domains. We compare and contrast our new norm map to the traditional Dedekind-Hasse norm. We prove that factoring in an almost Dedekind domain is in one-to-one correspondence to factoring in the new normset, improving upon this results in \cite {Coykendall}. In \cite {Grams}, an atomic almost Dedekind domain was constructed with a trivial Jacobson radical. We pursue atomicity in almost Dedekind domains with nonzero Jacobson radicals, showing the usefulness of the new norm we introduced. We state theorems with regard to specific classes of almost Dedekind domains. We provide a necessary condition for an almost Dedekind domain with nonzero Jacobson radical to be atomic.

Let $I$ be a square-free monomial ideal of a polynomial ring $R$ such that $\dim (R/I) = 2$. We give explicit formulas for computing the $a_i$-invariants $a_i(R/I^{(n)})$, $i=1,2$, and the Castelnuovo-Mumford regularity $\reg (R/I^{(n)})$ for all $n$. The values of these functions depend on the structure of an associated graph. It turns out that these functions are linear functions of $n$ for all $n \ge 2$.

Let $A$ be an Artinian Gorenstein algebra over an infinite field~$k$ of characteristic either 0 or greater than the socle degree of $A$. To every such algebra and a linear projection $\pi $ on its maximal ideal $\mathfrak {m}$ with range equal to the socle $\Soc (A)$ of $A$, one can associate a certain algebraic hypersurface $S_{\pi }\subset \mathfrak {m}$, which is the graph of a polynomial map $P_{\pi }:\ker \pi \to \Soc (A)\simeq k$. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras $A$, $\widetilde {A}$ are isomorphic if and only if any two hypersurfaces $S_{\pi }$ and $S_{\tilde {\pi }}$ arising from $A$ and $\widetilde {A}$, respectively, are affinely equivalent. The proof is indirect and relies on a geometric argument. In the present paper, we give a short algebraic proof of this statement. We also discuss a connection, established elsewhere, between the polynomials $P_{\pi }$ and Macaulay inverse systems.

Let $D$ be an integral domain with quotient field $K$ and $\Omega $ a finite subset of $D$. McQuillan proved that the ring $\Int (\Omega ,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega $, that is, $f\in K[X]$ such that $f(\Omega )\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring $\Int _K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose image over $S$ is contained in $A$. We show that the integral closure of $\Int _K(S,A)$ is equal to the contraction to $K[X]$ of $\Int (\Omega _S,D_F)$, for some finite subset $\Omega _S$ of integral elements over $D$ contained in an algebraic closure $\olK $ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega _S)$. Moreover, the integral closure of $\Int _K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form $D[X]+p(X)K[X]$, where $p(X)$ is a monic non-constant polynomial over $D$: we prove that the integral closure of such a pullback is equal to the ring of polynomials over $K$ which are integral-valued over the set of roots $\Omega _p$ of $p(X)$ in $\overline K$.