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- Ordered_ring abstract "In abstract algebra, an ordered ring is a commutative ring R with a total order ≤ such that for all a, b, and c in R: if a ≤ b then a + c ≤ b + c. if 0 ≤ a and 0 ≤ b then 0 ≤ ab.Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers. (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.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.If a is an element of an ordered ring R, then the absolute value of a, denoted |a|, is defined thus:where -a is the additive inverse of a and 0 is the additive identity element.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.".
- Ordered_ring wikiPageExternalLink document.pdf.
- Ordered_ring wikiPageID "525697".
- Ordered_ring wikiPageRevisionID "599879537".
- Ordered_ring hasPhotoCollection Ordered_ring.
- Ordered_ring subject Category:Ordered_groups.
- Ordered_ring subject Category:Real_algebraic_geometry.
- Ordered_ring type Abstraction100002137.
- Ordered_ring type Group100031264.
- Ordered_ring type OrderedGroups.
- Ordered_ring comment "In abstract algebra, an ordered ring is a commutative ring R with a total order ≤ such that for all a, b, and c in R: if a ≤ b then a + c ≤ b + c. if 0 ≤ a and 0 ≤ b then 0 ≤ ab.Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.".
- Ordered_ring label "Anillo ordenado".
- Ordered_ring label "Ordered ring".
- Ordered_ring label "Pierścień uporządkowany".
- Ordered_ring label "حلقة مرتبة".
- Ordered_ring sameAs Anillo_ordenado.
- Ordered_ring sameAs Pierścień_uporządkowany.
- Ordered_ring sameAs m.02llwt.
- Ordered_ring sameAs Q3782642.
- Ordered_ring sameAs Q3782642.
- Ordered_ring sameAs Ordered_ring.
- Ordered_ring wasDerivedFrom Ordered_ring?oldid=599879537.
- Ordered_ring isPrimaryTopicOf Ordered_ring.