From an abstract algebra point of view, why is zero so special? Every time I see a definition of some space or algebraic structure, I always see the exception: "division is defined for all elements of the set $F$ except for zero."

My question is, why all this discrimination against zero? In abstract algebra terms, zero is only the additive identity, and abstract algebra is supposed to abstract our notions of numbers, why is it that this impossibility to divide by zero does not dissolve once we elevate from our current preconceptions of numbers? Why is zero still so special even after all these generalizations?
 A: Let $F$ be some kind of algebraic object ($F$ is probably some kind of ring). Whilst $0$ may just be the additive identity when we consider $F$ as an additive group, once we add in multiplication to the equation, $0$ has another property: since we want multiplication and addition to be distributive, for any $a\in F$, we have
$$0\cdot a = (0+0)\cdot a = (0\cdot a) + (0\cdot a)$$so $$0\cdot a = 0.$$
Hence if we want our algebraic object to obey the usual distributive laws, multiplication by zero cannot be injective, and therefore we cannot expect it to have a well-defined inverse.
Zero is not actually special here - it is the property that multiplication by $0$ is not injective that is important. For example, if $R$ is a ring with non-zero elements $a,b$ with $$ab = 0$$ (e.g. in $\mathbb Z/4\mathbb Z$, $2\cdot 2=0$) then multiplication by $a$ in this ring is not injective, so we cannot hope to find an inverse of $a$ in some extension of $R$.
Rings with no such elements are special and are called Integral Domains. They have the property that they can be embedded into a field - i.e. we can find an extension of $R$ where every non-zero element has an inverse; the unique smallest field an integral domain $R$ can embed into is called its field of fractions.
