Ordered ring

partially ordered ring

A ring $ R $( not necessarily associative) which is a partially ordered group under addition and in which for any elements $ a , b , c \in R $ the inequalities $ a \leq b $ and $ c \geq 0 $ imply $ a c \leq b c $ and $ c a \leq c b $. Every ring is an ordered ring for the trivial order. As examples of ordered rings one may take an ordered field; the ring of real functions on a set $ X $, where $ f \leq g $ means that $ f ( x) \leq g ( x) $ for all $ x \in X $; or a matrix ring over an ordered ring $ R $, where, by definition, $ \| a _ {ij} \| \leq \| b _ {ij} \| $ if $ a _ {ij} \leq b _ {ij} $ for all $ i , j $. If $ R $ is an ordered ring, then the set

$$ P = \{ {x } : {x \in R , x \geq 0 } \} $$

is called its positive cone. The positive cone of an ordered ring completely defines the order: $ x \leq y $ if and only if $ y - x \in P $. A subset $ P $ of a ring $ R $ can serve as the positive cone for some order if and only if

$$ P \cap ( - P ) = \{ 0 \} ,\ \ P + P \subseteq P \ \textrm{ and } \ \ P P \subseteq P . $$

The equation $ P \cup ( - P ) = R $ is equivalent to the totality of the order (cf. Totally ordered set).

An ordered ring that is totally ordered or lattice-ordered is accordingly called a totally ordered or lattice-ordered ring (cf. also Archimedean ring). Lattice-ordered rings turn out to be distributive lattices, and their additive groups are torsion-free (cf. Lattice-ordered group). Certain questions in the theory of associative rings and, in particular, in the theory of radicals have analogues in associative lattice-ordered rings. The class of rings which allow a lattice-ordered ring structure is not axiomatizable. If $ a , b , c $ are elements of a lattice-ordered ring and $ c \geq 0 $, then the following relations hold:

$$ ( a \lor b ) c \geq a c \lor b c ,\ \ c ( a \lor b ) \geq c a \lor c b , $$

$$ ( a \wedge b ) c \leq a c \wedge b c ,\ c ( a \wedge b ) \leq c a \wedge c b . $$

Ideals in lattice-ordered rings which are convex subgroups (cf. Convex subgroup) of the additive group are called $ l $- ideals. The quotient ring by an $ l $- ideal can be made into a lattice-ordered ring in a natural way. The homomorphism theorem holds.

A lattice-ordered ring $ R $ is called a functional ring or an $ f $- ring if it satisfies any of the following equivalent conditions: 1) $ R $ is isomorphic to a lattice-ordered subring of a direct product of totally ordered rings; 2) for any $ a , b , x \in R $ one has the implication

$$ ( a \wedge b = 0 \textrm{ and } x \geq 0 ) \Rightarrow ( a \wedge b x = a \wedge x b = 0 ) ; $$

3) for any subset $ X $ of $ R $ the set

$$ \{ {y } : {y \in R , \forall x \in X x \wedge y = 0 } \} $$

is an $ l $- ideal; and 4) for any $ a , b \in R $,

$$ ( a \lor 0 ) ( b \lor 0 ) \wedge ( - a \lor 0 ) = ( b \lor 0 ) ( a \lor 0 ) \wedge ( - a \lor 0 ) = 0 . $$

Condition 4) shows that $ f $- rings form a variety of signature $ \{ + , - , 0 , \cdot , \lor , \wedge \} $. Neither of the equations in this condition is a consequence of the other. Not every $ f $- ring can be imbedded in an $ f $- ring with a unit element. If $ a , b , c $ are elements of an $ f $- ring and $ c > 0 $, then one has

$$ ( a \lor b ) c = a c \lor b c ,\ c ( a \lor b ) = c a \lor c b , $$

$$ ( a \wedge b ) c = a c \wedge b c ,\ c ( a \wedge b ) = c a \wedge c b , $$

$$ ( a \lor ( - a ) ) ( b \lor ( - b ) ) = a b \lor ( - a b ) ,\ a ^ {2} \geq 0 , $$

as well as the implication $ ( a \wedge b = 0 ) $ $ \Rightarrow $ $ ( a b = 0 ) $.

An order of an ordered ring $ R $ with a positive cone $ P $ can be extended to a total order such that $ R $ becomes a totally ordered ring if and only if for any finite set $ a _ {1} \dots a _ {n} $ in $ R $ one can choose $ \epsilon _ {i} = 1 $ or $ - 1 $ such that in the semi-ring generated by $ P $ and the elements $ \epsilon _ {1} a _ {1} \dots \epsilon _ {n} a _ {n} $ the sum of any two non-zero elements is non-zero. With $ P = \{ 0 \} $ one obtains a criterion for the possibility of having a total order on the ring.

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) MR1366860 MR0751233 MR1567177 MR0598630 MR0227053 MR0123490 MR0095754 MR0046339 MR0029876 MR0001959 MR1562917 Zbl 0537.06001 Zbl 0505.06001 Zbl 0198.33603 Zbl 0153.02501 Zbl 0126.03801 Zbl 0080.00412 Zbl 0049.01602 Zbl 0033.10103 Zbl 0063.00402 Zbl 0009.39402 Zbl 66.0100.04 Zbl 61.0997.10 Zbl 61.0127.03 Zbl 60.0113.10 Zbl 60.0093.01
[2]	A.A. Vinogradov, "The non-axiomatizability of lattice-ordered rings" Math. Notes , 21 (1977) pp. 253–254 Mat. Zametki , 21 : 4 (1977) pp. 449–452
[3]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) MR0171864 Zbl 0137.02001
[4]	A. Bigard, K. Keimel, S. Wolfenstein, "Groupes et anneaux reticulés" , Springer (1977) MR0552653 Zbl 0384.06022
[5]	G.W. Brumfiel, "Partially ordered rings and semi-algebraic geometry" , Cambridge Univ. Press (1979) MR0553280 Zbl 0415.13015
[6]	S.A. Steinberg, "Radical theory in lattice-ordered rings" Symp. Mat. Ist. Naz. Alta Mat. , 21 (1977) pp. 379–400 MR0472639 Zbl 0374.06012
[7]	S.A. Steinberg, "Examples of lattice-ordered rings" J. of Algebra , 72 : 1 (1981) pp. 223–236 MR0634624 Zbl 0485.06004

Comments

For a survey of the current state-of-the-art in the field see the second part of [a1].

References

[a1]	J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989) MR1094821 Zbl 0699.00020