On the Arens Homomorphism

Let $E$ be a unital $f$-module over an $f$-algebra $A$. With the help of Arens extension theory, a $(A^{\sim})_{n}^{\sim}$ module structure on $E^{\sim}$ can be defined. The paper deals mainly with properties of the Arens homomorphism $\eta\colon(A^{\sim})_{n}^{\sim}\to \operatorname{Orth}(E^{\sim})$, which is defined by the $(A^{\sim})_{n}^{\sim}$ module structure on $E^{\sim}$. Necessary and sufficient conditions for an $A$ submodule of $E$ to be an order ideal are obtained.