User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation

Theorem
Let $(R,+,\circ)$ be a ring.

Let $C$ be a cone in $R$ compatible with $(R,+,\circ)$.

Let $\mathcal R$ be the transitive relation on $R$ induced by $C$.

Then $\mathcal R$ is compatible with $(R,+,\circ)$

Proof
By the definition of a cone compatible with a ring and the fact that a cone compatible with a group operation induces a compatible relation, $\mathcal R$ is compatible with $+$.

We need to show that if $0 \mathrel{\mathcal R} x,y$ then $0 \mathrel{\mathcal R} x \circ y$

Suppose that $0 \mathrel{\mathcal R} x,y$.

Then $x = x+(-0) \in C$ and, similarly, $y \in C$.

Thus $x\circ y \in C$, so $x\circ y + (-0) \in C$, so $0 \mathrel{\mathcal R} x \circ y$