If $A$ is a C*-algebra, we say that a subset $I\subseteq A$ is hereditary if $$ 0\leq x \leq y \in I \Rightarrow x\in I. $$ It is is well known that closed 2-sided ideals are hereditary.

Would it also be true for arbitrary 2-sided ideals? What about self-adjoint 2-sided ideals?