Definition:Null Ring

Theorem
A ring with one element is called the null ring.

That is, the null ring is $\left({\left\{{0_R}\right\}, +, \circ}\right)$, where ring addition and the ring product are defined as:


 * $0_R + 0_R = 0_R$
 * $0_R \circ 0_R = 0_R$

The null ring is a trivial ring and therefore a commutative ring.

Consequently, a non-null ring is a ring with more than one element.

Proof
It needs proof that the null ring is actually a ring.

So, taking the ring axioms in turn:

A: Addition forms a Group
$\left({\left\{{0_R}\right\}, +}\right)$ is a group.

This follows from the definition of the Trivial Group: the element $0_R$ is the identity for the operation $+$.

M0: Closure of Ring Product
$\left({\left\{{0_R}\right\}, \circ}\right)$ is closed:
 * $0_R \circ 0_R = 0_R$

by definition.

M1: Associativity of Ring Product

 * $0_R \circ \left({0_R \circ 0_R}\right) = 0_R \circ 0_R = \left({0_R \circ 0_R}\right) \circ 0_R$

Thus $\circ$ is associative.

D: Distributivity of Ring Product over Addition
$\circ$ distributes over $+$ in $\left({\left\{{0_R}\right\}, +, \circ}\right)$:

First we have:
 * $0_R \circ \left({0_R + 0_R}\right) = 0_R$

by definition.

Then we have:

Now, the fact that the null ring is a trivial ring arises (obviously) from the fact that $0_R \circ 0_R = 0_R$.

Also known as
Some authors refer to this as the zero ring.

Others refer to it as the trivial ring, but this term has been defined differently elsewhere.