TITLE: Are those properties sufficient for defining a field?
QUESTION [6 upvotes]: Consider the following properties:

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

Are those enough for defining a field?

REPLY [5 votes]: 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.