Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $\mathrm {Ass}_R(M)$ and $\mathrm {Ass}_R(N)$ are disjoint, then $M\otimes _RN$ is isomorphic to direct sum of copies of $R/\mathfrak {a}$, where $\mathfrak {a}$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,\mathfrak {m},k)$ in terms of the homological dimensions of the modules linked to the syzygies of the residue field $k$. As a result we characterize a local ring $(R,\mathfrak {m},k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$.

Let $Q$ be a regular local ring of dimension $3$. We show how to trim a Gorenstein ideal in $Q$ to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra $P$ padded with a nonzero graded vector space on which $P_{\ge 1}$ acts trivially. We explicitly construct an infinite family of such rings.

In previous work, we introduced the notion of perinormality and showed how it lies in the greater context of commutative algebra, fitting as it does between the class of Krull domains and the class of seminormal ($R_1$) domains. Here we develop the concept further, using several pullback (i.e., gluing) constructions that yield perinormal domains. In doing so, we introduce the concepts of relative perinormality and fragility for ring extensions, which we believe to be of independent interest.

Let $T$ be a complete equicharacteristic local (Noetherian) UFD of dimension $3$ or greater. Assuming that $|T| = |T/\mathfrak m|$, where $\mathfrak m$ is the maximal ideal of $T$, we construct a local UFD $A$ whose completion is $T$ and whose formal fibers at height one prime ideals have prescribed dimension between zero and the dimension of the generic formal fiber. If, in addition, $T$ is regular and has characteristic zero, we can construct $A$ to be excellent.

Given a numerical semigroup $S$ and a positive integer $d$, the fraction $S/d=\{ x \in \mathbb {N} \mid dx \in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $S/d$ in terms of the minimal generators of $S$ and provide sharp upper bounds for the embedding dimension of $S/d$.

We introduce an order on the set of nondivisorial ideals of a numerical semigroup $S$, and link antichains of this order with the star operations on $S$; subsequently, we use this order to find estimates on the number of star operations on $S$. We then use them to find an asymptotic estimate on the number of nonsymmetric numerical semigroups with $n$ or less star operations, and to determine these semigroups explicitly when $n=10$.

Using an approach to the Jacobian conjecture by L.M. Druzkowski and K. Rusek, G. Gorni and G. Zampieri, and A.V. Yagzhev, we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert's problem in classical ring theory and the homogeneous dependence problem in affine algebraic geometry related to the Jacobian conjecture. We demonstrate these relations in concrete examples and formulate some open questions.