In this work, using the concept of multiple convex body semigroup, we present new families of Buchsbaum semigroups. We characterize Buchsbaum circle and convex polygonal semigroups and we describe algorithmic methods to check such characterizations.

Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The elasticity of $x$, denoted by $\rho \left(x\right)$, is the ratio of its largest factorization length to its shortest factorization length, and it measures how far $x$ is from having all its factorizations of the same length. The elasticity $\rho \left(M\right)$ of $M$ is the supremum of the elasticities of all non-unit elements of $M$. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of ${\mathbb{Q}}_{\ge 0}$). We characterize, in terms of the atoms, which Puiseux monoids $M$ have finite elasticity, giving a formula for $\rho \left(M\right)$ in this case. We also classify when $\rho \left(M\right)$ is achieved by an element of $M$. When $M$ is a primary Puiseux monoid (that is, a Puiseux monoid whose atoms have prime denominator), we describe the topology of the set of elasticities of $M$, including a characterization of when $M$ is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most $2$).

Let $R$ be an affine domain of dimension $n\ge 3$ over a field of characteristic $0$. Let $L$ be a projective $R\left[T\right]$-module of rank $1$ and $I\subset R\left[T\right]$ a local complete intersection ideal of height $n$. Assume that $I∕I^2$ is a surjective image of $L\oplus R{\left[T\right]}^{n-1}$. This paper examines under what conditions $I$ is a surjective image of a projective $R\left[T\right]$-module $P$ of rank $n$ with determinant $L$.

Let $\mathfrak{𝔮}$ denote an ideal of a local ring $\left(A,\mathfrak{𝔪}\right)$. For a system of elements $\underset{¯}{a}=a_1,\dots ,a_t$ such that $a_i\in {\mathfrak{𝔮}}^{c_i},i=1,\dots ,t$, and $n\in \mathbb{Z}$ we investigate a subcomplex and a factor complex of the Čech complex ${Č}_{\underset{¯}{a}}{\otimes }_{A}M$ for a finitely generated $A$-module $M$. We start with the inspection of these cohomology modules that approximate in a certain sense the local cohomology modules $H_{\underset{¯}{a}}^i\left(M\right)$ for all $i\in \mathbb{N}$. In the case of an $\mathfrak{𝔪}$-primary ideal $\underset{¯}{a}A$ we prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.

Let $R=\mathbb{𝕂}\left[a,b,c,d\right]∕\left(ad-bc\right)$ or $\mathbb{𝕂}\left[a,b,c\right]∕\left(ac-b^2\right)$, where $\mathbb{𝕂}$ is a field of arbitrary characteristic and $a,b,c,d$ are indeterminates. In this paper we will show that there is an order preserving embedding from the poset of Hilbert functions of homogeneous ideals of $R$ to the poset of homogeneous ideals of $R$. We also obtain related results for Betti numbers when the characteristic of $\mathbb{𝕂}$ is $0$.

Let $G$ be a graph obtained by taking $r\ge 2$ paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo–Mumford regularity of the quotient $S∕I\left(X\right)$, where $S$ is the polynomial ring on the edges of $G$ and $I\left(X\right)$ is the vanishing ideal of the projective toric subset parameterized by $G$. This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the $2$-connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo–Mumford regularity of $S∕I\left(X\right)$ with the combinatorial structure of $G$, for a general graph.

Let $G$ be a finite group. A finite unordered sequence $S=g_1\bullet \cdots \bullet g_{\ell }$ of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals $1_G$, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid $\mathcal{ℱ}\left(G\right)$ with basis $G$, and we study the submonoid $\mathcal{ℬ}\left(G\right)\subset \mathcal{ℱ}\left(G\right)$ of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if $G$ is abelian. In case of abelian groups, $\mathcal{ℬ}\left(G\right)$ is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of $\mathcal{ℬ}\left(G\right)$.

In this paper, we characterize Prüfer $v$-multiplication domains as integral domains which have the property that the existence of a generalized solution of any system of linear equations is equivalent to a weak equality of the determinantal ideals of the coefficient matrix and the augmented matrix of the system. In fact, we obtain a more general result for commutative rings of weak global $\tau$-dimension (in the sense of Bueso, Van Ostaeyen and Verschoren) at most one, where $\tau$ is a half-centered hereditary torsion theory.

Skolem properties are usually defined on rings of integer-valued polynomials, and are thought to be about finitely generated ideals. We contend that Skolem properties are more naturally stated in terms of $\star$-ideals. By this method, we characterize the classes of ideals on which $\text{ Int}\left(\mathbb{Z}\right)$ has the Skolem and strong Skolem properties. We extend the definitions of Skolem properties to rings comprising rational functions. We further define semistar operations which generalize the Skolem closure in this broader context. Finally we give some conditions under which a ring has Skolem properties.