The lattice of ideals of certain rings

Let $A$ be a unitary ring and let $(\mathbf{I}(\mathbf{A}),\subseteq )$ be the lattice of ideals of the ring $A\mathrm{.}$ In this article we will study the property of the lattice $(\mathbf{I}(\mathbf{A}),\subseteq )$ to be Noetherian or not, for various types of rings $A$. In the last section of the article we study certain rings that are not Boolean rings, but all their ideals are idempotent.