Positive Elements of Ordered Ring

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Theorem

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

Let $P$ be the set of positive elements of $R$ , that is, $P = R_{\ge 0}$.


Then:

$(1): \quad P + P \subseteq P$
$(2): \quad P \cap \paren {-P} = \set {0_R}$
$(3): \quad P \circ P \subseteq P$


If $\le$ is a total ordering, that is, if $\struct {R, +, \circ, \le}$ is a totally ordered ring, then:

$(4): \quad P \cup \paren {-P} = R$


The converse is also true:

Let $\struct {R, +, \circ}$ be a ring.

Let $P \subseteq R$ such that $(1)$, $(2)$ and $(3)$ are satisfied.


Then there is one and only one ordering $\le$ compatible with the ring structure of $R$ such that $P = R_{\ge 0}$.


Also, if $(4)$ is also satisfied, then $\le$ is a total ordering.


Proof

Necessary Condition

First, suppose that $\le$ is compatible with the ring structure of $R$.

Hence:

$(OR1): \quad \le$ is compatible with $+$
$(OR2): \quad \forall x, y \in R: 0_R \le x, 0_R \le y \implies 0_R \le x \circ y$.


$(1)$: Let $x, y \in R: 0_R \le x, 0_R \le y$.

Then $0_R + 0_R \le x + y$ by the fact that $\le$ is compatible with $+$.

Thus $0_R \le x + y$ and thus $x + y \in P$.


$(2)$: By Properties of Ordered Ring item 4, $-P = \set {x \in R: x \le 0_R}$.

Let $x \in P \cap \paren {-P}$.

Then $x \le 0_R$ and $0_R \le x$.

So from the antisymmetric nature of $\le$, $x = 0_R$.


$(3)$: This is equivalent to $0_R \le x, 0_R \le y \implies 0_R \le x \circ y$ which is one of the properties of being compatible with the ring structure of $R$.


$(4)$: Now if $\le$ is a total ordering, then $\forall x \in R$, either $x \le 0_R$ or $0_R \le x$, and the result follows.


Sufficient Condition

Let $P \subseteq R$ such that $(1)$, $(2)$ and $(3)$ are satisfied.

By item $(OR2)$ of Properties of Ordered Ring, we have:

$x \le y \iff 0 \le y + \paren {-x}$

so there is at most one ordering on $R$ compatible with the ring structure of $R$ such that $P = R_{\ge 0}$, namely, the one that satisfies:

$x \le y \iff y + \left({-x}\right) \in P$

Now we need to show that $\le$ thus defined has the required properties.

Reflexivity

By $(2)$:

$\forall x \in R: x \le x$ because $x + \paren {-x} = 0_R \in P$


Antisymmetry

Let $x \le y$ and $y \le x$.

Then $y + \paren {-x} \in P$ and $-\paren {y + \paren {-x} } = x + \paren {-y} \in P$.

Thus by $(2)$, $y + \paren {-x} = 0_R$ and thus $y = x$.


Transitivity

If $x \le y$ and $y \le z$, then $y + \paren {-x} \in P$ and $z + \paren {-y} \in P$.

But as $z + \paren {-x} = z + \paren {-y} + y + \paren {-x}$, we have that $z + \paren {-x} \in P$ from $(1)$.

Hence $x \le z$.


Let $x \le y$.

Then $z + x \le z + y$ since $\paren {z + y} + \paren {-\paren {x + x} } = y + \paren {-x} \in P$.


Finally, Ordering Compatible with Ring holds because of $(3)$.


If $(4)$ holds, then $\forall x, y \in R$, either $y + \paren {-x} \in P$ or $x + \paren {-y} = -\paren {y + \paren {-x} } \in P$, that is, either $x \le y$ or $y \le x$.

$\blacksquare$


Sources