Last Updated：2026/01/01
Sentence
The
positive
elements
in
an
ordered
ring
allow
us
to
compare
elements
to
0,
but
we
know
in
the
integers
that
we
can
compare
any
two
elements
to
each
other.
For
example,
we
know
that
4>2
because
4-2>0.
We
can
extend
this
idea
to
any
ordered
ring.
If
R
is
an
ordered
ring
and
a,b∈R,
then
we
know
by
trichotomy
that
exactly
one
of
the
following
must
be
true:
a-b>0,
a-b=0,
or
-(a-b)>0.
Quizzes for review
The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that 4>2 because 4-2>0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b∈R, then we know by trichotomy that exactly one of the following must be true: a-b>0, a-b=0, or -(a-b)>0.
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