Ideal theory in $f$-algebras

Abstract

The paper deals mainly with the theory of algebra ideals and order ideals in /-algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete /-algebra with unit element every algebra ideal is an order ideal iff the /-algebra is normal. This result is based on the fact that the range of every orthomorphism in a uniformly complete normal Riesz space is an order ideal.