Let $S$ be a principal ideal domain. Recall that a Laurent polynomial algebra over $S$ is an $S$-algebra of the form $S[T_{1},\ldots, T_{n},T_{1}^{-1}, \ldots, T_{n}^{-1}]$. Generalizing this notion, we call an $S$-algebra of the form $S[T_{1},\dots, T_{n},f_{1}^{-1}, \ldots, f_{n}^{-1}]$ a quasi Laurent polynomial algebra in $n$ variables over $S$ if $T_{1},\ldots, T_{n}$ are algebraically independent over $S$ and $f_{i}=a_{i}T_{i}+b_{i}$, where $a_{i} \in S \backslash 0$ and $b_{i} \in S$ are such that $(a_{i}, b_{i})S=S$, for each $i=1, \ldots,n$. It has been shown recently that a locally Laurent polynomial algebra in $n$ variables over $S$ is itself a Laurent polynomial algebra. Now suppose $A$ is a locally quasi Laurent polynomial algebra in $n$ variables over $S$. In this note, we investigate the question: `is $A$ necessarily quasi Laurent polynomial in $n$ variables over $S$?' We first give a sufficient condition for the question to have an affirmative answer. Moreover, when $S$ is semi-local with two maximal ideals and contains the field of rationals $\mathbf{Q}$, we give examples of $S$-algebras which are locally quasi Laurent polynomial in two variables but not quasi Laurent polynomial in two variables.

One introduces a class of projective parameterizations that resemble generalized de~Jonqui\`eres maps. Any such parametrization defines a birational map $\mathfrak{F}$ of $\pp^n$ onto a hypersurface $V(F)\subset \pp^{n+1}$ with a strong handle to implicitization. From this side, the theory developed here extends recent work of Ben\'{\i}tez and D'Andrea on monoid parameterizations. The paper deals with both the ideal theoretic and effective aspects of the problem. The ring theoretic development gives information on the Castelnuovo-Mumford regularity of the base ideal of $\mathfrak{F}$. From the effective side, we give an explicit formula of $\deg(F)$ involving data from the inverse map of $\mathfrak{F}$ and show how the present parametrization relates to monoid parameterizations.

We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then tables exist with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all fibers with strictly positive marginals are connected by the quadratic moves that correspond to conditional independence statements. The property persists under natural operations such as gluing along cliques, but we also construct examples of graphical models not enjoying this property. We also provide a negative answer to a question of Engstr\"om, Kahle and Sullivant by demonstrating that the global Markov ideal of the complete bipartite graph $K_{3,3}$ is not radical.

Our analysis of the positive margins property depends on computing the primary decomposition of the associated conditional independence ideal. The main technical results of the paper are primary decompositions of the conditional independence ideals of graphical models of the $N$-cycle and the complete bipartite graph $K_{2,N-2}$, with various restrictions on the size of the nodes.

We introduce and study ``quasidualizing" modules. An artinian $R$-module $T$ is \emph{quasidualizing} if the homothety map $\comp R\rightarrow\hom TT$ is an isomorphism and $\Ext{i}{T}{T}=0$ for each integer $i>0$. Quasidualizing modules are associated to semidualizing modules via Matlis duality. %We study the properties of %quasidualizing modules. An $R$-module $L$ is \emph{derived $T$-reflexive} if %the map $L\rightarrow\hom{\hom LT}{T}$ is an isomorphism %and $\Ext{i}{L}{T}=0=\Ext{i}{\hom LT}{T}$ for each integer $i>0$. We investigate the associations via Matlis duality between subclasses of the Auslander class and Bass class and subclasses of derived $T$-reflexive modules.

In this note we give a characterization of the rings with finitely many minimal prime ideals.

Toric face rings are a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e and Herzog et al., graded Betti numbers of $k$ over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by R\"{o}mer about the sufficient condition for the Koszul property. The conjecture is inspired by Fr\"{o}berg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.

We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of Pragacz and Weyman, who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal varieties are irreducible representations for a certain classical Lie algebra.

We show that the co-chordal cover number of a graph $G$ gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.