TITLE: What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
QUESTION [8 upvotes]: The title pretty much says it all.
Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period."  In my experience, mathematical definitions are rarely completely arbitrary.  Therefore I figure there must be a good reason for insisting that a field be a non-trivial ring (among other things), but it's not obvious to me.  I realize that a trivial ring would make for a very boring field, but it is also a rather boring ring, and yet it is not disallowed as such.
EDIT: I found a hint of a rationale in the statement that "the zero ring ... does not behave like a finite field," but I could not find in the source for this assertion exactly how the zero ring fails to behave like a finite field.

REPLY [14 votes]: I think of the requirement $0\neq1$ for fields as a consequence of the desires for (1) fields to be integral domains and (2) integral domains to satisfy $0\neq1$. Of course that just shifts the question to: Why do I want integral domains to have $0\neq1$?  That desire comes from my (quite general) inclination to treat the empty set just like other finite sets. Here's how that's relevant: The key clause in the definition of integral domains (besides the axioms for commutative rings with unit) is that if $xy=0$ then $x=0$ or $y=0$. It follows immediately by induction that, if a product of $n\geq 2$ factors is $0$, then there must be a $0$ among those factors. The same holds trivially for $n=1$ under the obvious convention that the product of a single factor is that factor.  So I'd like it to hold also for $n=0$.  Now the product of no factors is $1$, so what I want is that, if $1=0$, then there is a $0$ in the (empty) set of factors.  As there is no $0$ (or anything else) in the empty set, I infer that $1\neq0$.
(The same underlying idea explains why I don't regard the integer $1$ as prime and why I want lattices to have top and bottom elements.)