For a polynomial map $\tupBold{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \tupBold{f}$, where $h: k^m \to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic~$0$ and $\tupBold{f}$ is surjective, we will show that $g = h \circ \tupBold{f}$ implies that $h$ is a polynomial.

Let $S_1$ and $S_2$ be two affine semigroups, and let $S$ be the gluing of $S_1$ and $S_2$. Several invariants of $S$ are related to those of $S_1$ and $S_2$; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given.

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding ``whiskers'' to graphs. In this paper, we study a similar construction for building a simplicial complex $\Delta_\chi$ from a coloring $\chi$ of a subset of the vertices of $\Delta$ and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes `\textit{regular crossings} (r.c.) \textit{monomial}.' We prove that r.c.~monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension~2 r.c.~monomial centers.

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

We study the relationship between algebras generated by idempotents over a commutative ring $R$ with identity and algebras that are quotient rings of group algebras $RG$ for torsion abelian groups $G$ without an element whose order is a zero-divisor in $R$. The main purpose is to seek conditions for $R$ to hold the equality between these two kinds of algebras.

In this note, we present several new results on derivations in characteristic $p\neq0$, together with a Jacobian identity that we recently discovered through a miscalculation. Our main identity states that, if $k$ is a field of characteristic $p$ and $f_{1},\ldots,f_{n}$ belong to the polynomial ring $k\left[x_{1},\ldots,x_{n}\right]$ and $J(f)$ equals the determinant of the $n\times n$ Jacobian matrix, $[\partial f_{i}/\partial x_{j}]$, then {\footnotesize \[ \sum^{p-1}_{i_{1}=1}\!\!\cdots\!\!\sum^{p-1}_{i_{n}=1}\!\!f_{1}^{i_{1}}\!\!\cdots \! f_{n}^{i_{n}}\nabla\!\left(\!f_{1}^{p-1-i_{1}}\!\cdots f_{n}^{p-1-i_{n}}\!\right)\!=\!\left(-1\right)^{n}\left(J\left(f\right)\right)^{p-1}\!, \]} where $\nabla=\partial^{n(p-1)}/\partial x_{1}^{p-1}\cdots\partial x_{n}^{p-1}$. We conclude with a brief discussion of nilpotent derivations in characteristic $p$ in connection with the degree less than $p$ version of the Jacobian conjecture.

We show there exist no nontrivial semidualizing modules for nonmodular rings of invariants of order $p^n$ with $p$ a prime.

Let $I$ be the edge ideal of a clutter $\calC$ in a polynomial ring $S$. In this paper, we present estimations of the Stanley depth of $I$ as well as the Stanley regularity of $S/I$, in terms of combinatorial data from the clutter~$\calC$.

Given a finite purely inseparable extension of positive characteristic, this paper determines necessary and sufficient conditions on the ring of relative differential operators to establish whether the extension is modular.