ordered ring

In an ordered ring, no negative element is a square.

The definition of absolute value given for real numbers above can be extended to any ordered ring.

It follows that Z together with the above ordering is an ordered ring.

The complex numbers do not form an ordered ring (or ordered field).

This property is sometimes used to define ordered rings instead of the second property in the definition above.

An ordered ring that is not trivial is infinite.

The integers are a discrete ordered ring, but the rational numbers are not.

Suppose is a commutative ordered ring, and .

Ordered rings are familiar from arithmetic.

But suppose they do ask questions -" "Tell them all you know," ordered Ring.

This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.

Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order.

Ordered groups, ordered rings and ordered fields have algebraic structure compatible with an order on the set.

She paused, then added, "I also ordered rings made for Surreal and Wilhelmina.

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

In abstract algebra, an ordered ring is a commutative ring with a total order such that for all a, b, and c in R:

During the maturation of osteoclast precursors, groups of podosomes form higher ordered ring structures which ultimately coalesce into a band about the cell periphery.

IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings.

Every ordered ring is an f-ring, so every subdirect union of ordered rings is also an f-ring.

Schwartz, Niels; Madden, James J. Semi-algebraic function rings and reflectors of partially ordered rings.

The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains).

The above definition requires well-defined operations of addition, multiplication and comparison, therefore the notion of a linear inequality may be extended to ordered rings, in, particular, to ordered fields.

An ordered ring, also called a totally ordered ring, is a partially ordered ring where is additionally a total order.

A discrete ordered ring or discretely ordered ring is an ordered ring in which there is no element between 0 and 1.

For example, an Archimedean partially ordered ring is a partially ordered ring where 's partially ordered additive group is Archimedean.