In this paper, we exhibit examples of four-dimensional seminormal domains $A$ and $B$ which are finitely generated over the field $\mathbb{C} $ (or $\mathbb{R} $) such that $A[X,Y] \cong B[X,Y]$ but $A[X]\ncong B[X]$.

The core of an ideal $I$ is the intersection of all reductions of $I$. We prove that the core behaves well under extension to the trivial extension. Also, we describe the core as a colon ideal of a power of any reduction and a power $I$, for a class of ideals $I$ in Cohen-Macaulay rings.

Via the theory of reverse lexicographic squarefree initial ideals of toric ideals, we give a new class of Gorenstein Fano polytopes (reflexive polytopes) arising from a pair of stable set polytopes of perfect graphs.

Let $R$ be a ring. Following the literature, $R$ is called \textit {residually finite} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $R/I_r$ is finite. In this note, we define a \textit {strictly} infinite commutative ring $R$ with identity to be \textit {residually small} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $|R/I_r|\lt |R|$. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.

This paper describes a construction of a minimal free resolution of a generic ideal defined by determinants in characteristic zero. It produces not only the free modules in the resolution, but it also defines the maps between them explicitly and in detail in terms of idempotents in the group algebra of the symmetric group.

Let $\mathbb{k} $ be a field, and let $X$ and $Y$ be two locally noetherian $\mathbb{k} $-schemes (respectively, $\mathbb{k} $-formal schemes) with dualizing complexes $R_X$ and $R_Y$, respectively. We show that $R_X \boxtimes _{\mathbb{k} } R_Y$ (respectively, its derived completion) is a dualizing complex over $X\times _{\mathbb{k} } Y$ if and only if $X\times _{\mathbb{k} } Y$ is locally noetherian of finite Krull dimension.

In this paper, we give a toric representation of the associated ring of a polyomino which is obtained by removing a convex polyomino from its ambient rectangle.

The unitary Cayley graph of a ring $R$ is the simple graph whose vertices are the elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x-y$ is a unit in $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $g$ such that $G$ can be embedded into an orientable surface $\mathbb {S}_{g}$. It is proven that, for a given positive integer $g$, there are at most finitely many finite commutative rings whose unitary Cayley graphs have genus $g$. We determine all finite commutative rings whose unitary Cayley graphs have genus 1, 2 and 3, respectively.

In this paper, we study the two different topics related to sequentially Cohen-Macaulay modules. Questions arise as to when the sequentially Cohen-Macaulay property preserves the localization and the module-finite extension of rings.