I am trying to get my head around essential ideals. In literature I found 2 definitions:

An ideal $I$ in a C\*-algebra $A$ is essential in $A$

(i) if $aI = 0$ implies $a=0$, $a\in A$; or

(ii) if every other non-zero ideal in $A$ has a non-zero intersection with $I$.

I'm trying to show that they are equivalent. While I was able to show (i) implies (ii) by proving the contrapositive, I'm struggling with showing (ii) implies (i). 

Also, is it in general true that if $I$ is an essential ideal in a C\*-algebra $A$ and $J$ is an essential ideal in a C\*-algebra $B$ then $I \otimes J$ is an essential ideal in $A \otimes B$, where $\otimes$ denotes the minimal tensor product, and how can I prove this?

I am very grateful to anyone who can help with these questions.