A remark on the spectra of rings with Gabriel dimension
If A is a commutative ring with identity, denote by \({\mathcal I}(A)\) the lattice of ideals of A and by Spec(A) the topological space (endowed with the Zariski topology) of prime ideals of A. The aim of this short note is to prove the following result: Let A and B two commutative rings with identity such that there exists an isomorphism \({\mathcal I}(A)\cong {\mathcal I}(B)\) of partially ordered sets. If A has a Gabriel dimension, then the topological spaces Spec(A) and Spec(B) are homeomorphic. This extends a theorem established in a more complicated manner by \textit{C. Năstăsescu} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 19, 87-92 (1974; Zbl 0289.13001)] in the case A is Noetherian.
scientific article; zbMATH DE number 3880841