I use following definitions from Corso di geometria-Stoka Marius-CEDAM-1995:
let be $(C,+)$ a commutative group and $(C,\cdot)$ a semigroup $$(C,+,\cdot) \text{ is ring iff }\begin{cases} \forall x\in C, \forall y\in C, \forall z\in C: (x+y)\cdot z=(x\cdot z)+(x\cdot y)\\ \forall x\in C, \forall y\in C, \forall z\in C: x\cdot (y+ z)=(x\cdot y)+(x\cdot z) \end{cases}$$
let be $(C,+,\cdot)$ a ring $$(C,+,\cdot) \text{ is commutative ring iff } \forall x \in C, \forall y\in C: (x\cdot y)=(y\cdot x) $$
let be $(C,+,\cdot)$ a ring $$(C,+,\cdot) \text{ is ring with 1 iff } \exists x=:1 \in C,\forall y \in C: (x \cdot y)=y=(y\cdot x) $$
I thinked, are the following definitions of skew field and field possible?
let be $(C,+,\cdot)$ a ring with 1 $$(C,+,\cdot) \text{ is skew field iff } \forall x \in C\setminus{0},\exists y \in C: (x \cdot y)=1=(y \cdot x)$$
let be $(C,+,\cdot)$ a skew field $$(C,+,\cdot) \text{ is field iff } \forall x \in C,\forall y \in C: (x \cdot y)=(y\cdot x)$$
If yes, is $(\{0\}, +, \cdot)$ a field?