TITLE: Proper definition of a field?
QUESTION [7 upvotes]: I just read a very short definition of a field where it was said that a field is a set of elements $K$ with two maps from the field into the field itself, such that
$K$ is an abelian group with $+$. 
$K \backslash \{0\}$ is an abelian group with $\cdot$.
Furthermore, we have the distributive law $x(y+z)=xy+xz$.
Is this already sufficient to show that $0x=0$?
If anything is unclear, please let me know.

REPLY [1 votes]: You can show that $0=x-x=x(1-1)=x0$ which is the other way around than what you wanted. To get $0x=0$ I think you will need to add that multiplication is commutative (which I believe you need to properly define a field) or alternatively that multiplication is also right distibutive. Another (short) definition of a field would be a commutative ring so that the non-zero elements form a group under multiplication.
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