Let $D$ be an integral domain with the quotient field $K$, $X$ an indeterminate over $K$ and $x$ an element of $D$. The Bhargava ring over $D$ at $x$ is defined to be ${𝔹}_x\left(D\right):=\{f\in K\left[X\right]:\text{for}\text{all}a\in D,f(xX+a)\in D\left[X\right]\}$. In fact, ${𝔹}_x\left(D\right)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of ${𝔹}_x\left(D\right)$ under localization. In particular, we prove that ${𝔹}_x\left(D\right)$ behaves well under localization at prime ideals of $D$, when $D$ is a locally finite intersection of localizations. We also attempt a classification of integral domains $D$...

The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex $Ho{m}_{R̂}((,R̂),M)$ is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.

Let $(R,𝔪)$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq {(x_1,...,x_n)}^2$ is a graded ideal of a polynomial ring $S=K[x_1,...,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ of the first syzygy module ${\mathrm{Syz}}_1\left(𝔪\right)$ of $𝔪$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.