Let $R$ be a commutative Noetherian ring of dimension~$d$ and \[ B=R[X_1,\ldots ,X_m,Y_1^{\pm 1},\ldots ,Y_n^{\pm 1}] \] a Laurent polynomial ring over $R$. If $A=B[Y,f^{-1}]$ for some $f\in R[Y]$, then we prove the following results: (i) if $f$ is a monic polynomial, then the Serre dimension of $A$ is $\leq d$. The case $n=0$ is due to Bhatwadekar, without the condition that $f$ is a monic polynomial. (ii) The $p$th Euler class group $E^p(A)$ of $A$, defined by Bhatwadekar and Sridharan, is trivial for $p\geq \max \{d+1,\dim A -p+3\}$. The case $m=n=0$ is due to Mandal and Parker.

The notion of Ulrich ideals was introduced by Goto et al.~\cite {GOTWY}. They developed an interesting theory on Ulrich ideals. In particular, they gave a characterization of Ulrich ideals of Gorenstein numerical semigroup rings that are generated by monomials. Using this result, in this paper, we investigate Ulrich ideals of Gorenstein numerical semigroup rings with embedding dimension~3 that are generated by monomials. In particular, we completely determine when such Ulrich ideals are existent in those rings.

We introduce the concept of s-Hankel hypermatrix, which generalizes both\nHankel matrices and generic hypermatrices. We study two determinantal ideals\nassociated to an s-Hankel hypermatrix: the ideal I<s,t> generated by certain 2\nx 2 slice minors, and the ideal \\tilde{I}<s,t> generated by certain 2 x 2\ngeneralized minors. We describe the structure of these two ideals, with\nparticular attention to the primary decomposition of I<s,t>, and provide the\nexplicit list of minimal primes for large values of s. Finally we give some\ngeometrical interpretations and generalise a theorem of Watanabe.\n

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 $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ 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 ${\rm Int}_K(S,A)$ 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 ${\rm Int}_K(S,A)$ is equal to the contraction to $K[X]$ of ${\rm Int}(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elements over $D$ contained in an algebraic closure $\bar K$ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of ${\rm Int}_K(S,A)$ 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 $\bar K$.

Let $D$ be an integrally closed domain, $S(D)$ the set of star operations on $D$, $w$ the $w$-operation, and $S_w(D) = \{* \in S(D) \mid w \leq *\}$. Let $X$ be an indeterminate over $D$ and $N_v = \{f \in D[X] \mid c(f)_v = D\}$. In this paper, we show that, if $D$ is a Pr\ufer $v$-multiplication domain (P$v$MD), then $|S_w(D)| = |S_w(D[X])| = |S(D[X]_{N_v})|$. We prove that $D$ is a P$v$MD if and only if $|\{* \in S_w(D) \mid *$ is of finite type$\}|\lt \infty$. We then use these results to give a complete characterization of integrally closed domains $D$ with $|S_w(D)| \lt \infty$.

Some interesting properties of almost Cohen-Macaulay rings are investigated, and a Serre type property connected with this class of rings is studied.

In this paper we study certain determinantal ideals that extend the class of ideals of Herzog-Northcott type introduced by O'Carroll and Planas-Vilanova. As is well known, in a three-dimensional Cohen-Macaulay local ring, the second symbolic powers of ideals of Herzog-Northcott type can be controlled well. We aim to generalize this fact considering ``saturation" instead of ``symbolic power." Furthermore, in order to compare the saturation with the symbolic power, we study the associated primes of the powers of certain determinantal ideals.

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are identical on the graded Cohen-Macaulay modules over a graded ring which is of graded finite representation type and representation directed.

We determine the defining equations of the Rees algebra of an ideal $I$ in the case where $I$ is a square-free monomial ideal such that each connected component of the line graph of the hypergraph corresponding to $I$ has at most $5$ vertices. Moreover, we show in this case that the non-linear equations arise from even closed walks of the line graph, and we also give a description of the defining ideal of the toric ring when $I$ is generated by square-free monomials of the same degree. Furthermore, we provide a new class of ideals of linear type. We show that when $I$ is a square-free monomial ideal with any number of generators and the line graph of the hypergraph corresponding to $I$ is the graph of a disjoint union of trees and graphs with a unique odd cycle, then $I$ is an ideal of linear type.