A field is a nonzero commutative ring ...

Question

I was confused when I read this statement. I thought a Ring must have the additive inverse $0$. Does the statement imply that there is no zero in a field?

Answer 1

All rings have $0$, but a non-zero ring $R$ is not the zero ring {${0}$}.

Answer 2

All rings have $0$. Nonzero ring means not the trivial ring, the ring with one element. [Not to be confused with the so-called "field" with one element, which is a phrase borne of poetic license.]

Answer 3

The "zero ring" is the trivial ring $R = \{0\}$. On the other hand, a non-zero ring is one such that $|R|>1$, i.e., it has an element $0$ together with some other non-zero elements (which of course must necessarily include "$1$", the multiplicative identity).