Are those properties sufficient for defining a field?

Question

Consider the following properties:

1. $a(x+y) = ax + ay$
2. $x + y = y + x$
3. $ax = xa$
4. $x + 0 = x$
5. $x \cdot 1 = x$
6. for every $x\ne 0$ there's a $y$ such that $xy=1$.

Are those enough for defining a field?

Answer 1

These axioms do not require the existence of the opposite (for addition). The set of the not negative rational numbers with the usual operations satisfies these axioms and is not a field.

Answer 2

No. You still need additive inverses and the two associative properties. As a quick counterexample, consider the set of all nonnegative integers reals $S=\{x\in\mathbb{R}\mid x\ge0\}$. They satisfy all six properties. But it's not a field because there are no additive inverses.