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$\begingroup$

i) If $a$ is positive and $b$ is negative, then their product $ab$ is a negative element.

ii) And if $a \neq b$ are positive elements and $a \preceq b$, then $b-a$ is a positive element.

Thanks for your help.

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    Sorry, $R$ not $\mathbb{R}$. – 2017-02-28
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    What did you try? What properties does an ordered ring have? – 2017-02-28
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    I know that in any ordered ring, (1) If $a$ is a positive element, then its negative (opposite) $−a$ is a negative element, and vice versa. (2) If $c$ is a negative element and $a ⪯ b$, then $bc ⪯ ac$. (3) The product of two positive or two negative elements is positive. (4) The identity $1$ is a positive element. – 2017-02-28
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    Hint: For problem (i), try using property (2). For problem (ii), I'm guessing you have some property that's similar to "if $a \leq b$, then $a+c \leq b+c$" which would be helpful. – 2017-02-28
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    For (i), would I say that if $a$ is positive, then $0 ⪯ a$; and if $b$ is negative, $b ⪯ 0$? Then, could I say $a\cdot b=(a)\cdot (-a)$? – 2017-02-28
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    @SebastianAlexis I wouldn't say $a\cdot b = (a)\cdot (-a)$ unless you know that $b = -a$ or something of the sort. But you know that $0\preceq a$ and $b$ is negative. What does property (2) say about $ab$? – 2017-02-28

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