Diagonal Relation on Ring is Ordering Compatible with Ring Structure

Theorem
Let $\struct {R, +, \circ, \preceq}$ be a ring whose zero is $0_R$.

Then the diagonal relation $\Delta_R$ on $R$ is an ordering compatible with the ring structure of $R$.

Proof
From Diagonal Relation is Ordering and Equivalence, we have that $\Delta_R$ is actually an ordering on $R$.

From the definition of the diagonal relation:
 * $\tuple {x, y} \in \Delta_R \iff x = y$

Thus:

Similarly:

So $\Delta_R$ is compatible with $+$.

Then note that:

Hence the result, from the definition of an ordering compatible with the ring structure of $R$.