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 $D$...

For $k\in \mathbb{N}\cup \left\{\infty \right\}$ and $U$ open in $ℝ$, let ${\mathrm{C}}^k\left(U\right)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f\in {\mathrm{C}}^k\left(U\right)$ there is $g\in {\mathrm{C}}^{\infty }\left(ℝ\right)$ with $U\subseteq \mathrm{coz}g$ and $h\in {\mathrm{C}}^k\left(ℝ\right)$ with ${fg|}_U{=h|}_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring ${\mathrm{C}}^k\left(ℝ\right)$ are deduced from this, in particular that ${\mathrm{Q}}_{\mathrm{cl}}\left({\mathrm{C}}^k\left(ℝ\right)\right)=\mathrm{Q}\left({\mathrm{C}}^k\left(ℝ\right)\right)$.

In this paper, we will present several necessary and sufficient conditions on a commutative ring such that the algebraic and geometric local cohomologies are equivalent.