Let $R$ be a ring and $\leq$ be a partial order. $R$ is partially ordered by $\leq$ if
For any $a$, $b$, $x\in R$,
\[a\leq b \enspace\rimp\enspace a+x\leq b+x.\]For any $a$, $b\in R$,
\[0\leq a \,\land\, 0\leq b \enspace\rimp\enspace 0\leq ab.\]$R$ is totally ordered if $\leq$ is total.