Definition:Trivial Ring

Theorem
A ring $$\left({R, +, \circ}\right)$$ is a trivial ring iff:


 * $$\forall x, y \in R: x \circ y = 0_R$$

A trivial ring is a commutative ring.

Proof
To prove that a trivial ring is actually a ring in the first place, we need to check the ring axioms for a trivial ring $$\left({R, +, \circ}\right)$$:

Taking the ring axioms in turn:

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

This follows from the definition.

M0: Closure of Ring Product
$$\left({R, \circ}\right)$$ is closed:

From Ring Product with Zero, we have $$x \circ y = 0_R \in R$$.

M1: Associativity of Ring Product
$$\circ$$ is associative on $$\left({R, +, \circ}\right)$$:


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

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


 * $$x \circ \left({y + z}\right) = 0_R$$ by definition.

Then:

$$ $$

... and the same for $$\left({y + z}\right) \circ x$$.

The fact of its commutativity follows from $$x \circ y = 0_R = y \circ x$$.