Order-to-ring ideals via submultiplicative lattice seminorms in lattice-ordered algebras

Abstract

This paper characterizes lattice-ordered algebras with the order-to-ring ideal property, i.e., those in which every order ideal is a ring ideal. Our main result shows that an Archimedean lattice-ordered algebra has this property if and only if it is an f-algebra admitting a nil-faithful submultiplicative lattice seminorm, where nil-faithful means that elements with zero seminorm are nilpotent. As a consequence, we prove that an Archimedean semiprime f-algebra has the order-to-ring ideal property if and only if it admits a submultiplicative lattice norm. These results correct and extend earlier incorrect work in the literature and yield a necessary and sufficient condition for all orthomorphisms on an Archimedean vector lattice to be central.