The prime spectrum and dimension of ideal transform algebras

Abstract

Let $(R,\mathfrak{m} )$ be a commutative Noetherian local ring of dimension $d\geq 1$, and let $I$ be a non-nilpotent ideal of $R$ such that the ideal transform functor $D_I(-)$ is exact. In this paper, it is shown that the finitely generated flat $R$-algebra $D_I(R)$ is a Noetherian ring of dimension $n=\dim R/\Gamma _I(R)-1$. Also, it is shown that, under Zariski topologies on the sets $Spec D_I(R)$ and $Spec R/\Gamma _I(R)$, there is a homeomorphism of topological spaces: \[ \widetilde {\eta ^*}:Spec D_I(R)\longrightarrow Spec R/\Gamma _I(R)\setminus V((I+\Gamma _I(R))/\Gamma _I(R)). \]