Allowing the zero element in a field to have an inverse In the definition of a field one of the required properties is that
every element other than zero has a multiplicative inverse.
It's vague whether the zero is forced not to have an inverse or not, as in is the property
every element other than zero has a multiplicative inverse and zero does not
or
every element other than zero has a multiplicative inverse and zero may or may not?
Regardless, what differences would occur if we adopted one definition as opposed to the other?
 A: The fact that $0$ has no inverse in a ring (with identity $1\ne0$) is a result of the distributive property of multiplication and mentioning it in a definition is redundant:
$$\quad ab=(0+a)b=0b+ab \implies 0b=ab-ab=0$$
A: If you kept all of the other usual axioms, then
$$ 0 = 0 \cdot 0^{-1} = 1 $$
$$ x = x \cdot 1 = x \cdot 0 = 0 $$
and so all numbers are zero. This is rather degenerate. This is called the "zero ring"; usually we do not consider this ring to be a field.
A: If $0$ had a multiplicative inverse, then $0=0*0^{-1}=1$, so your ring would be the trivial one. By the way, this holds for any ring, not just for fields.
A: If in a ring $R$ the zero $0$ has an inverse, then $R=0$ (in fact, $1 = 0 \cdot 0^{-1} = 0$, hence $r = 1r = 0r = 0$ for all $r \in R$). One wants fields to be non-zero for many reasons, for example for the uniquenes of the dimension (remark that $R^n \cong R^m$ for all $n,m$ when $R=0$). Therefore the second definition is correct (i.e. coincides with the usual definition of a field), whereas the first and the third also include the zero ring, which is not a field.
