$F$ is a field $\implies$ $1_F \neq 0_F$

Question

I came across a step in a proof which stated :

$F$ is a field $\implies$ $1_F \neq 0_F$ .

but isn't it in general true even if it isn't a field...

Answer 1

In the definition of a field $F$, it is stated that it has a multiplicative identity $1 \in F \setminus \{0\}$, however in the definition of a general ring $R$, it is only stated that there exist $1 \in R$ such that $1 \dot a = a, \ \forall \ a \in R$. You see that this property holds for $R = \{0\}$ with $1=0$, so in a general ring we might not have $1 \neq 0$. However as said above, in a field $F$ we have $1 \in F \setminus \{0\}$ by definition, so your implication holds.

See for example: http://en.wikipedia.org/wiki/Field_(mathematics), where it is said in the defintion that $1 \in F \setminus \{0\}$.

In writing this answer, I'm refering to the definitions stated in my textbook(s). You may have a different definition in your textbook ?