Residuated Lattices with Noetherian Spectrum

Abstract

In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion of i-Noetherian residuated lattice is introduced and related properties are investigated. We proved that a residuated lattice is i-Noetherian iff every ideal is principal. Moreover, we show that a residuated lattice has the spectrum of a Noetherian space iff it is i-Noetherian.

Keywords

• Noetherian residuated lattice
• ideal
• prime ideal
• Noetherian spectrum
• ring of algebraic integers
• Bezout rings