Did I define a field correctly? Let $F$ be a set, $\forall a, b, c \in F$, with two binary operations, addition and multiplication, defined on $F$, i.e. $(F, +, \times )$. Then $F$ is a field iff the following axioms hold:


*

*Closure:
1.1. Addition: $\forall a,b \in F, \exists a + b \in F$.
1.2. Multiplication: $\forall a,b \in F, \exists a \times b \in F$.

*Associativity:
2.1. Addition: $\forall a,b,c \in F, (a + b) + c = a + (b + c)$.
2.2. Multiplication: $\forall a,b,c \in F, (a \times b)\times c=a \times (b \times c)$.

*Identity element:
3.1. Addition: $\exists e_{Additive} \in F, s.t. \forall a \in F, a + e_{Additive} = e_{Additive} + a= a$.
(Edit - a correction, see comments).
3.2. Multiplication: $\exists e_{Multiplicative} \in F, s.t. \forall a \in F, a \times e_{Multiplicative} = e_{Multiplicative} \times a= a$.
(Edit - a correction, see comments).

*Inverse element:
4.1. Addition: $\exists b, e_{Additive} \in F, \forall a \in F, s.t. a + e_{Additive} = a, a + b = b + a = e_{Additive}$.
(Edit - defined $e_{Additive}$)
4.2. Multiplication: $\exists b, e_{Multiplicative} \in F, \forall a \in F$, s.t. $a \times e_{Multiplicative} = a, a \times b = b \times a = e_{Multiplicative}$.
(Edit - defined $e_{Multiplicative}$)

*Commutativity:
5.1. Addition: $\forall a, b \in F, a + b = b + a$.
5.2. Multiplication: $\forall a, b \in F, a \times b = b \times a$.

*Distributivity: $\forall a, b, c \in F, a \times (b + c) = a \times b + a \times c = b \times a + c \times a = (b + c) \times a$.

*The additive and multiplicative identity elements are different: $\exists e_{Additive}, e_{Multiplicative} \in F, \forall a \in F$, s.t. $a + e_{Additive} = a, a \times e_{Multiplicative} = a, e_{Additive} \neq e_{Multiplicative}$.
(Edit - axiom 7 added (hopefully correctly formulated), see comments and answers.)

I want to know if the definition above is correct, if not why not, and if my use of quantifiers is correct, if not why not to avoid false statements and the such. Now, I know fields and whatnot are defined on wikipedia and other sources, including here, over a million times but I want to use "my own" ordering of these axioms, if that makes any sense, and ask the following questions.


*

*Are all of these axioms detailed enough to define a field? E.g. In $(6)$ I didn't specify left or right distributivity but it is implied because of $(5.2)$, if I am not mistaken or is it even required to do so. Did I miss anything like that?

*Using the numbering of the axioms above, if they are stated correctly, which axioms must hold for a structure to be a ring? Now, I am not enrolled in Abstract Algebra or such so please don't dive in too deep if possible.
Any help much appreciated, thank you.
 A: More succinctly, a field is a commutative ring in which every nonzero element is a unit.
A: To sum up comments:
To your question 1: Your definition is correct when amended as follows:


*

*(crucial:) Add an axiom 7: $e_{add} \neq e_{mult}$.

*(formal:) In no.4 (and the amended no.7), somehow make clear that $e_{add}$ resp. $e_{mult}$ refer to the elements defined in no.3.

*(optional:): In nos. 3,4 and 6, amend a "left" and a "right" version. As long as axiom 5 is in place, this is of course redundant, but it will help to clarify an answer to your second question.


To your question 2:
For a ring, leave out 3.2, 4.2, 5.2 and the amended no.7. For a unital ring (which however often is tacitly assumed even if people just write "ring"), leave out nos. 4.2, 5.2, and 7, but keep 3.2. (in a left and a right version). For a commutative ring (which is almost always understood to be unital as well, and we'll follow that), leave out nos. 4.2 and 7.
By the way, if you leave out just 5.2 but keep everything else (including no.7 and with left and right versions where applicable (although some of them might turn out to be redundant), you get what is called a division ring or skew field.
