Product of Positive Element and Element Greater than One

Theorem

Let $\struct {R, +, \circ, \le}$ be an ordered ring with unity $1_R$ and zero $0_R$.

Let $x, y \in R$.

Suppose that $x > 0_R$ and $y > 1_R$.

Then $x \circ y > x$ and $y \circ x > x$.

Proof

\(\ds y\)

\(>\)

\(\ds 1_R\)

\(\ds x \circ y\)

\(>\)

\(\ds x \circ 1_R\)

$x > 0_R$, Properties of Ordered Ring: $(6)$

\(\ds x \circ y\)

\(>\)

\(\ds x\)

Definition of Unity of Ring

A similar argument shows that $y \circ x > x$.

$\blacksquare$