A note on the ring $R[X]_B$

Let $R$ be a commutative ring with identity, $R[X]$ the polynomial ring over $R$ and
$$B = \{f \in R[X] \,|\, {\rm \ the \ coefficient \ of \ the \ least \ degree \ term \ of \ } f {\rm \ is \ } 1 \}.$$
Then $B$ is a multiplicative subset of $R[X]$; so we obtain the quotient ring $R[X]_B$ of $R$ by $B$. Also, since $B$ is a subset of $N=\{f \in R[X] \,|\, c(f) = R\}$, $R[X]_B$ is a subring of the Nagata ring $R[X]_N$ of $R$. In this talk, we investigate the PIR property and Prüfer domain property in $R[X]_B$. For this purpose, we examine the ideal theory, dimension theory and factorization theory in $R[X]_B$.