# 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

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[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 |

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Ordered ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ordered_ring&oldid=48068