Edit: In a semiring $(R, +, \cdot, 0, 1)$, $0$ and $1$ are additive identity and multiplicative identity, respectively such that $0$ is multiplicatively absorbing, that is $0\cdot a=0=a\cdot 0$ for all $a\in R$ and $1$ is additively absorbing (in particular), that is $1+a=1=a+1$.
I doubt $1$ is also called additive zero? Please correct me if i am wrong.
$0$ is multiplicatively absorbing in a ring, which can easily be shown. For instance, let $R$ be a ring and $a\in R.$ Then since $0$ is an additive identity, $a0=a(0+0)=a0+a0\Rightarrow a0=0.$ Similarly, $0a=0.$
However, the additive identity $0$ is not additively absorbing. That would mean that $\forall a\in R,$ $0+a=a+0=0,$ which is obviously false.
In a ring, $0$ is just notation for the additive identity, so it doesn't have to be $0.$ It could be $1.$ The term "additive zero" is much less conventional than "additive identity," even though $0$ is often referred to as the additive identity. The multiplicative identity or unity is often written as $1,$ though it may not always be $1.$ Consider the ring $(R,+,\cdot)$ where $\forall a,b\in R, a+b := a+b-1 \wedge a\cdot b := a+b-ab.$ The additive identity is $1,$ not $0,$ and the multiplicative identity is $0,$ not $1.$