Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals

Abstract

We describe the ideals of an idealization $D⋉M$ by triples $(I,N,\phi )$, where $I$ is an ideal of $D$, $N$ is a submodule of $M$ with $IM\subseteq N$, and $\phi \in {\mathrm{Hom}}_D(I,M/N)$. For a domain $D$ and torsion-free $M$, the nonzero proper ideals of $D⋉M$ with nonzero annihilator are exactly $0⋉N$. As an application, for $R=\mathbb{R}[[t]]⋉\mathbb{R}[[t]]$, the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality $2^{{\aleph }_0}$. This answers negatively a question of Behboodi and Rakeei.