Partially ordered ring

In abstract algebra, a partially ordered ring is a ring (A, +, · ), together with a compatible partial order, i.e. a partial order on the underlying set A that is compatible with the ring operations in the sense that it satisfies:



and imply that

for all .[1] Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring where 's partially ordered additive group is Archimedean.[2]

An ordered ring, also called a totally ordered ring, is a partially ordered ring where is additionally a total order.[1][2]

An l-ring, or lattice-ordered ring, is a partially ordered ring where is additionally a lattice order.


The additive group of a partially ordered ring is always a partially ordered group.

The set of non-negative elements of a partially ordered ring (the set of elements x for which , also called the positive cone of the ring) is closed under addition and multiplication, i.e., if P is the set of non-negative elements of a partially ordered ring, then , and . Furthermore, .

The mapping of the compatible partial order on a ring A to the set of its non-negative elements is one-to-one;[1] that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.

If S is a subset of a ring A, and:

then the relation where iff defines a compatible partial order on A (ie. is a partially ordered ring).[2]

In any l-ring, the absolute value of an element x can be defined to be , where denotes the maximal element. For any x and y,



An f-ring, or PierceBirkhoff ring, is a lattice-ordered ring in which [4] and imply that for all . They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is negative, even though being a square.[2] The additional hypothesis required of f-rings eliminates this possibility.


Let X be a Hausdorff space, and be the space of all continuous, real-valued functions on X. is an Archimedean f-ring with 1 under the following point-wise operations:


From an algebraic point of view the rings are fairly rigid. For example, localisations, residue rings or limits of rings of the form are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings, is the class of real closed rings.


A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.[3]

in an f-ring.[3]

The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.[5]

Every ordered ring is an f-ring, so every subdirect union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a subdirect union of ordered rings.[2] Some mathematicians take this to be the definition of an f-ring.[3]

Formally verified results for commutative ordered rings

IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the ring1 context.[6]

Suppose is a commutative ordered ring, and . Then:

The additive group of A is an ordered group OrdRing_ZF_1_L4
iff OrdRing_ZF_1_L7
and imply
x is either in the positive set, equal to 0, or in minus the positive set. OrdRing_ZF_3_L2
The set of positive elements of is closed under multiplication iff A has no zero divisors. OrdRing_ZF_3_L3
If A is non-trivial (), then it is infinite. ord_ring_infinite


  1. Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7.
  2. Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389.
  3. Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 2330, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8.
  4. denotes infimum.
  5. Hager, Anthony W.; Jorge Martinez (2002). "Functorial rings of quotientsIII: The maximum in Archimedean f-rings". Journal of Pure and Applied Algebra. 169: 51–69. doi:10.1016/S0022-4049(01)00060-3.
  6. "IsarMathLib" (PDF). Retrieved 2009-03-31.

Further reading

  • Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69.
  • Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp
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