Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle f.a,F\rangle$ for every $a\in A,f\in A^*, F\in A^{**}$, and define $\langle f,F\square G\rangle=\langle G*f,F\rangle$, for all $F,G\in A^{**}, f\in A^{*}$. the product $\square$ on $A^{**}$ is called first Arens product and it makes $A^{**}$ into a Banach algebra.
Also we say that the Banach algebra $A$ is semi-prime if for all ideals $I$ of $A$, the $I^2=\{0\}$ implies $I=\{0\}$.
Question1: Under which conditions, $A$ is a left or right or bi-ideal of $(A^{**},\square)$?
Question2: If $A$ is semi-prime Banach algebra, could we say that $A^{**}$ is semiprime? If not, under which conditions we could say it?