The lattice of ideals of certain rings

Let A be a unitary ring and let \((\mathbf {I(A)},\subseteq )\) $(\mathbf{I}(\mathbf{A}),\subseteq )$ be the lattice of ideals of the ring A. In this article, we will study the property of the lattice \((\mathbf {I(A)},\subseteq )\) $(\mathbf{I}(\mathbf{A}),\subseteq )$ 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, not fields, but all their ideals are idempotent.