This is an ambiguity in abstract algebra texts I keep coming across. A perfect example is in defining the "zero divisor". 
How am I supposed to take the "0"? Because authors choose to use the familiar symbols from the real numbers, it conflates the concepts unnecessarily.
I dont know if "0" refers to additive identities, or if its referring to multiplicative absorption elements. Are these seemingly disparate notions intimately related in the most abstract sense?
To be clear, since we are dealing with multi-operator structures (fields and rings), there are two different operators I can interpret the 0 through. I was never taught that an additive identity was the same as a multiplicative absorption element; I just know they happen to be the same thing in the real numbers but I dont know this to be a general truth for rings.
Best Answer
Yes, the additive identity in a general ring is always a multiplicative absorbing element.
Consider an arbitrary ring $R$, with additive identity $0$. For any $a\in R$, we can see that $$a\cdot 0 = a \cdot (0 + 0) = a\cdot 0 + a \cdot 0$$
Thus we have shown that $a\cdot 0 = a\cdot 0 + a\cdot 0$. We know by closure that $a\cdot 0 = b$ for some $b\in R$, so we can add the additive inverse of $b$ to both sides of our equation. We denote the additive inverse of $b$ in the usual fashion by $-b$. Thus we conclude that $a\cdot 0 - b = a\cdot 0 + a\cdot 0 - b$, so indeed $0 = a\cdot 0$, thus $0$ is an absorbing element as required.