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

 * First we prove that the null ring is actually a ring.

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

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

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:

$$ $$

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