Unitization of a lattice ordered ring with a truncation

Abstract

Let R be a lattice ordered ring along with a truncation in the sense of Ball. We give a necessary and sufficient condition on R for its unitization R ⊕ Q. to be again a lattice ordered ring. Also, we shall see that R ⊕ Q is a lattice ordered ring for at most one truncation. Particular attention will be paid to the Archimedean case. More precisely, we shall identify the unique truncation on an Archimedean ℓ-ring R which makes R ⊕ Q into a lattice ordered ring.