The right ideals of an alternative ring

It is proved that if $P$ is a right ideal and $I$ a two-sided ideal of an alternative ring $A$, then neither $P^2$ nor $IP$ is in general a right ideal of $A$. Moreover, it is shown that in the alternative ring $A$ the right annihilator of the right ideal $P$, i.e., the set $\mathfrak{Z}_r(P)=\{z\in A\mid Pz=0\}$, is not necessarily either a left or a right ideal, nor even a subring of $A$.