Example of a communtative ring with two operations where the identity elements are not distinct? I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity,  associativity, identity & inverse elements, but there was a remark in the axioms that said the identity element of addition must be distinct from then identity element of multiplication, $ 0 \neq 1$. Is there a reason why this must be, and is there an example of a ring where this is not the case?
 A: The identity element under addition is denoted by $0$. It has the property that $0*a = 0 = a*0$ for all $a\in R$.
The identity element under multiplication is denoted $1$. It has the property that $1*a = a = a*1$ for all $a \in R$.
If $0 = 1$, then for all $a \in R$, 
$$ 0 = 0*a = 1*a = a.$$
So $R = \{0\}$. We don't like to call this a field as it doesn't behave like one.
A: If $0 = 1$, then for any $x$ in the ring:
$$x = x.1 = x.0 = 0$$
and therefore there is only one element in the ring.
A: The zero ring, a ring containing only one element, necessarily has $0 = 1$, since this ring satisfies all the ring axioms (except $0 \neq 1$), but only has one element.  The zero ring is a bit pathological, so rather than writing "Let $R$ be any ring other than the zero ring", we exclude the zero ring via the "$0 \neq 1$" condition.
There is also a notion "field with one element".  Note that even if one works in a setting where the field with one element makes sense, this object is not a field, for the same axiomatic reason as is used to exclude the zero ring.
