In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R:^{[1]}

if a ≤ b then a + c ≤ b + c.

if 0 ≤ a and 0 ≤ b then 0 ≤ ab.
Contents

Examples 1

Positive elements 2

Absolute value 3

Discrete ordered rings 4

Basic properties 5

Notes 6
Examples
Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.^{[2]} (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.
Positive elements
In analogy with the real numbers, we call an element c ≠ 0 of an ordered ring positive if 0 ≤ c, and negative if c ≤ 0. The element c = 0 is considered to be neither positive nor negative.
The set of positive elements of an ordered ring R is often denoted by R_{+}. An alternative notation, favored in some disciplines, is to use R_{+} for the set of nonnegative elements, and R_{++} for the set of positive elements.
Absolute value
If a is an element of an ordered ring R, then the absolute value of a, denoted a, is defined thus:

a := \begin{cases} a, & \mbox{if } 0 \leq a, \\ a, & \mbox{otherwise}, \end{cases}
where a is the additive inverse of a and 0 is the additive identity element.
Discrete ordered rings
A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not.
Basic properties
For all a, b and c in R:

If a ≤ b and 0 ≤ c, then ac ≤ bc.^{[3]} This property is sometimes used to define ordered rings instead of the second property in the definition above.

ab = a b.^{[4]}

An ordered ring that is not trivial is infinite.^{[5]}

Exactly one of the following is true: a is positive, a is positive, or a = 0.^{[6]} This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.

An ordered ring R has no zero divisors if and only if the positive ring elements are closed under multiplication (i.e. if a and b are positive, then so is ab).^{[7]}

In an ordered ring, no negative element is a square.^{[8]} This is because if a ≠ 0 and a = b^{2} then b ≠ 0 and a = (b )^{2}; as either b or b is positive, a must be positive.
Notes
The list below includes references to theorems formally verified by the IsarMathLib project.

^

^ *

^ OrdRing_ZF_1_L9

^ OrdRing_ZF_2_L5

^ ord_ring_infinite

^ OrdRing_ZF_3_L2, see also OrdGroup_decomp

^ OrdRing_ZF_3_L3

^ OrdRing_ZF_1_L12
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.