ordered ring
التعريفات والمعاني
== English ==
=== Noun ===
ordered ring (plural ordered rings)
(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.
1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If
≤
{\displaystyle \leq }
is an ordering on
A
{\displaystyle A}
compatible with its ring structure, we shall say that
(
A
,
+
,
⋅
,
≤
)
{\displaystyle (A,\ +,\ \cdot ,\leq )}
is an ordered ring. An element
x
{\displaystyle x}
of an ordered ring
A
{\displaystyle A}
is positive if
x
≥
0
{\displaystyle x\geq 0}
, and
x
{\displaystyle x}
is strictly positive if
x
>
0
{\displaystyle x>0}
.
The set of all positive elements of an ordered ring
A
{\displaystyle A}
is denoted by
A
+
{\displaystyle A_{+}}
, and the set of all strictly positive elements of
A
{\displaystyle A}
is denoted by
A
+
∗
{\displaystyle A_{+}^{*}}
.
If
(
A
,
+
,
⋅
,
≤
)
{\displaystyle (A,\ +,\ \cdot ,\leq )}
is an ordered ring and if
≤
{\displaystyle \leq }
is a total ordering, we shall, of course, call
(
A
,
+
,
⋅
,
≤
)
{\displaystyle (A,\ +,\ \cdot ,\leq )}
a totally ordered ring; if
(
A
,
+
,
⋅
)
{\displaystyle (A,\ +,\ \cdot )}
is a field, we shall call
(
A
,
+
,
⋅
,
≤
)
{\displaystyle (A,\ +,\ \cdot ,\leq )}
an ordered field, and if, moreover,
≤
{\displaystyle \leq }
is a total ordering, we shal call
(
A
,
+
⋅
,
≤
)
{\displaystyle (A,\ +\ \cdot ,\leq )}
a totally ordered field.
1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19,
DEFINITION 1. — Given a commutative ring
A
{\displaystyle A}
, we say that an ordering on
A
{\displaystyle A}
is compatible with the ring structure on
A
{\displaystyle A}
if it is compatible with the additive group structure of
A
{\displaystyle A}
, and if it satisfies the following axiom:
(OR) The relations
x
≥
0
{\displaystyle x\geq 0}
and
y
≥
0
{\displaystyle y\geq 0}
imply
x
y
≥
0
{\displaystyle xy\geq 0}
.
The ring
A
{\displaystyle A}
, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings
Q
{\displaystyle \mathbb {Q} }
and
Z
{\displaystyle \mathbb {Z} }
, with the usual orderings, are ordered rings.
2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring
A
E
{\displaystyle A^{E}}
of mappings from a set
E
{\displaystyle E}
to an ordered ring
A
{\displaystyle A}
is an ordered ring.
3) A subring of an ordered ring, with the induced ordering, is an ordered ring.
(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.
2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77,
Definition 3.5.4. A ring
R
{\displaystyle R}
is an ordered ring if there exists a distinguished set
R
+
{\displaystyle R^{+}}
,
R
+
⊂
R
{\displaystyle R^{+}\subset R}
, called the set of positive elements, with the properties that:
(1) The set
R
+
{\displaystyle R^{+}}
is closed under addition and multiplication.
(2) If
x
∈
R
{\displaystyle x\in R}
then exactly one of the following is true: (trichotomy law)
(a)
x
=
0
{\displaystyle x=0}
,
(b)
x
∈
R
+
{\displaystyle x\in R^{+}}
,
(c)
−
x
∈
R
+
{\displaystyle -x\in R^{+}}
.
If further
R
{\displaystyle R}
is an integral domain we call
R
{\displaystyle R}
an ordered integral domain.
[…]
Lemma 3.5.9. If
R
{\displaystyle R}
is an ordered ring and
a
∈
R
{\displaystyle a\in R}
is a positive element, then the set
{
n
a
:
n
∈
N
}
⊂
R
+
{\displaystyle \{na:n\in \mathbb {N} \}\subset R^{+}}
.
[…]
Theorem 3.5.2. An ordered ring must be infinite.
==== Usage notes ====
While the ring is, strictly speaking, not necessarily associative or commutative, it may be defined as either or both by authors working within an overarching theory.
The property
if
a
≤
b
and
0
≤
c
then
c
a
≤
c
b
and
a
c
≤
b
c
{\displaystyle {\textsf {if}}\ a\leq b\ {\textsf {and}}\ 0\leq c\ {\textsf {then}}\ ca\leq cb\ {\textsf {and}}\ ac\leq bc}
in the definition is sometimes replaced by the equivalent
if
0
≤
a
and
0
≤
b
then
0
≤
a
b
{\displaystyle {\textsf {if}}\ 0\leq a\ {\textsf {and}}\ 0\leq b\ {\textsf {then}}\ 0\leq ab}
.
The order is said to be compatible with the ring structure of
R
{\displaystyle R}
(in the sense that order is preserved by addition and, to an extent, multiplication).
A partial order
≤
{\displaystyle \leq }
is a total order if and only if the trichotomy condition holds: in other words,
P
∪
−
P
=
R
{\displaystyle P\cup -P=R}
, where
P
=
{
x
:
x
∈
R
,
0
≤
x
}
{\displaystyle P=\left\{x:x\in R,0\leq x\right\}}
is the positive cone of
R
{\displaystyle R}
and
−
P
=
{
−
x
:
x
∈
P
}
{\displaystyle -P=\left\{-x:x\in P\right\}}
.
Consequently, in the total order case, it makes sense to define an absolute value applicable to every element of
R
{\displaystyle R}
:
|
x
|
=
{
x
,
if
0
≤
x
−
x
,
if
0
>
x
.
{\displaystyle |x|=\left\{{\begin{array}{rl}x,&{\textsf {if}}\ 0\leq x\\-x,&{\textsf {if}}\ 0>x.\end{array}}\right.}
==== Synonyms ====
(ring equipped with a partial order): partially ordered ring
(ring equipped with a total order): totally ordered ring
==== Hyponyms ====
(both senses): discrete ordered ring (= discretely ordered ring), ordered field
==== Derived terms ====
lattice-ordered ring
==== Translations ====
=== See also ===
ordered field
=== Further reading ===
Linearly ordered group on Wikipedia.Wikipedia
Ordered ring on Encyclopedia of Mathematics