About a definition of skew field and field, with example $\{0\}$ 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?
A skew-field (or skew field) is a ring in which the equations ax=b and ya=b with a≠0 are uniquely solvable. In the case of an associative ring (cf. Associative rings and algebras) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations ax=1 and ya=1 for any a≠0. A commutative associative skew-field is called a field. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form

(a−bb¯a¯)
over the field of complex numbers with the usual operations (see Quaternion). An example of a non-associative skew-field is the Cayley–Dickson algebra, consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see Alternative rings and algebras). Any skew-field is a division algebra either over the field of rational numbers or over a field of residues Fp=Z/(p). The skew-field of quaternions is a 4-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is 8-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [Ad], and also Topological ring). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see [Po]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [Sk], [He]). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional [ZhSlShSh]. A similar result applies to Mal'tsev skew-fields [Fi] (see Mal'tsev algebra) and to Jordan skew-fields [Ze] (see Jordan algebra). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see Imbedding of rings).

Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre
A skew-field (or skew field) is a ring in which the equations ax=b and ya=b with a≠0 are uniquely solvable. In the case of an associative ring (cf. Associative rings and algebras) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations ax=1 and ya=1 for any a≠0. A commutative associative skew-field is called a field. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form

(a−bb¯a¯)
over the field of complex numbers with the usual operations (see Quaternion). An example of a non-associative skew-field is the Cayley–Dickson algebra, consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see Alternative rings and algebras). Any skew-field is a division algebra either over the field of rational numbers or over a field of residues Fp=Z/(p). The skew-field of quaternions is a 4-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is 8-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [Ad], and also Topological ring). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see [Po]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [Sk], [He]). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional [ZhSlShSh]. A similar result applies to Mal'tsev skew-fields [Fi] (see Mal'tsev algebra) and to Jordan skew-fields [Ze] (see Jordan algebra). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see Imbedding of rings).

Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre